Is There a Number Bigger Than a Googolplex?

A googolplex is staggeringly large, but it is not even close to the ceiling of numbers. You can always produce a bigger number by adding one, multiplying, or raising it to a higher power. Mathematicians have named specific numbers so far beyond a googolplex that a googolplex looks like a rounding error by comparison, and beyond those named giants lie functions that grow so fast they cannot even be computed by any possible program.

What a Googolplex Actually Is

A googol is 10 raised to the power of 100, which means a 1 followed by 100 zeros. A googolplex takes that a step further: it is 10 raised to the power of a googol, or 10 to the 10 to the 100. Written out in full decimal notation, a googolplex would be a 1 followed by a googol zeros. You could not write that number down even if you used every particle in the observable universe as an ink dot for each digit. The universe contains roughly 10 to the 80 particles, so you’d run out of matter long before you finished.

The term was coined in 1938 by Milton Sirotta, the nine-year-old nephew of mathematician Edward Kasner, and it has since become the go-to example when people want to talk about an absurdly large number. Google’s name is a deliberate play on “googol,” a nod to the company’s ambition to organize a seemingly limitless amount of information. But despite its cultural fame, a googolplex occupies a surprisingly modest rung on the ladder of large numbers that mathematicians have studied.

The Simplest Way to Get Something Bigger

The most obvious answer to the title question is almost disappointingly simple: a googolplex plus one is bigger than a googolplex. So is a googolplex times two, or a googolplex squared. There is no largest finite number, because you can always perform another arithmetic operation and land on something higher. The natural numbers stretch on without end, and a googolplex is just one point along that infinite line.

You can also stack the operation that created the googolplex in the first place. A googolplex is 10 to the power of (10 to the 100). Raise 10 to the power of a googolplex and you get a “googolplexplex,” a number with a googolplex digits. You could keep stacking exponents forever, building a tower of powers that grows with each new layer. This kind of iterated exponentiation is called tetration, and it’s just the first step beyond ordinary exponents. Mathematicians have formalized operations that go beyond tetration as well, each one producing growth rates that make the previous operation look flat.

Named Numbers That Dwarf a Googolplex

Several specific numbers that arise in real mathematical proofs make a googolplex look vanishingly small. The most famous is probably Graham’s number, which appeared in a proof in combinatorics by Ronald Graham in the 1970s. Graham’s number is so large that the entire observable universe could not contain a conventional decimal representation of it, and even the number of digits in the number of digits (and so on, repeated billions of times) would still be too large to write. Ordinary exponentiation and even towers of exponents cannot get anywhere near it. To express Graham’s number you need an arrow notation that describes hyper-operations, each one incomprehensibly more powerful than the last, applied repeatedly.

Then there is TREE(3), which comes from a theorem in combinatorics about sequences of trees. TREE(3) is the length of the longest possible sequence of labeled trees with at most three labels that obeys a particular embedding rule. Its exact value is unknown, but it is finite and it vastly exceeds Graham’s number. Even the lower-level cousin of this function, the weak tree function, hints at the explosive growth involved: a recent constructive proof showed that the weak tree function evaluated at just 3 already exceeds 844 billion.

1arXiv. A Lower Bound for Kruskal’s Weak Tree Function: tree(3)≥844,424,930,131,960

To put that in perspective, the weak tree function is considered tame compared to the full TREE function. TREE(3) is so large that no tower of exponents, no matter how tall, can express it. Mathematicians working on these problems have shown that even for small inputs, tree-related functions display growth that blows past anything standard arithmetic can describe.

Fast-Growing Hierarchies

Mathematicians do not just stumble on large numbers by accident. There is a systematic framework for generating ever-larger functions, known as fast-growing hierarchies. These hierarchies start with familiar operations and iteratively build new ones, each level producing growth that dwarfs the one before. Multiplication is repeated addition. Exponentiation is repeated multiplication. Tetration is repeated exponentiation. Each step up this ladder creates functions that accelerate far more quickly than the previous tier.

Fast-growing hierarchies formalize this idea by indexing each level with a mathematical label called an ordinal, allowing the system to extend far beyond the handful of operations that have everyday names. Researchers have shown that these hierarchies represent the canonical way to discuss large finite numbers in proof theory, establishing a rigorous bridge between how we talk about big numbers and how we talk about the structure of mathematical proofs themselves.

2Cambridge University Press / Forum of Mathematics, Sigma. Functorial Fast-Growing Hierarchies

The practical upshot is that whenever someone invents a clever new way to produce a big number, mathematicians can usually locate it within this hierarchy and say precisely how fast it grows relative to other known functions. Graham’s number, for instance, sits at a relatively low level in these hierarchies. TREE(3) lives enormously higher. And the hierarchy itself never ends, so there is always room for something bigger.

The Busy Beaver Function and Uncomputability

All the numbers discussed so far, no matter how large, are in principle computable. Given enough time and memory, a computer could calculate them. But there exists a function that grows faster than any computable function ever could: the Busy Beaver function.

The Busy Beaver function asks a deceptively simple question. Consider all possible programs of a certain size. Among those programs that eventually halt (stop running), which one produces the most output? The answer for programs of size n is called Σ(n), the Busy Beaver value for n. For tiny inputs, the values are known: Σ(1) is 1, Σ(2) is 4, Σ(3) is 6, Σ(4) is 13. But the function explodes. Σ(5) was only recently pinned down, and Σ(6) is already beyond current mathematical knowledge.

3ACM SIGACT News. The Busy Beaver Frontier

What makes the Busy Beaver function special is not just its size but its fundamental uncomputability. No algorithm can compute Σ(n) for all n, and this limitation has been proven to be equivalent to the undecidability of the halting problem, one of the deepest results in theoretical computer science.

4ACM SIGACT News. A bound on the shift function in terms of the Busy Beaver function

This means the Busy Beaver function eventually outgrows every computable function, including every fast-growing hierarchy level that can be described by a computer program. It outgrows Graham’s number, TREE(3), and anything else you can define with a finite algorithm. A googolplex is not just smaller than the Busy Beaver function at large inputs; it is smaller by an amount that no computable process can bridge. The function has captivated mathematicians and computer scientists for decades precisely because it sits at the boundary of what is knowable.

3ACM SIGACT News. The Busy Beaver Frontier

Why Our Brains Struggle with Numbers This Large

If you find it hard to wrap your head around numbers beyond a googolplex, you’re in good company. Human intuition is systematically bad at grasping rapid growth. Research on what psychologists call exponential-growth bias shows that people consistently underestimate the results of compounding processes. In studies measuring how well people perceive exponential growth, participants on average perceived only about a third of the true impact of compounding, treating exponential curves almost as if they were straight lines.

5Journal of Economic Psychology. Exponential-growth bias and overconfidence

And exponential growth is only the beginning of the story. A googolplex is already an exponential tower two levels deep. Graham’s number involves operations that iterate far beyond exponentiation, and the Busy Beaver function grows faster than any computable process. If people reliably underestimate even simple compound interest, there is essentially no hope of intuitively grasping the magnitude of these numbers. The best anyone can do is understand the relationships between them: this function grows faster than that one, this number sits higher on the hierarchy than that one. The actual magnitudes involved are simply beyond any sensory or spatial metaphor.

This cognitive limitation helps explain why a googolplex feels like it ought to be “the biggest number.” It is already far beyond anything a person encounters in daily life, or even in most scientific contexts. The number of atoms in the universe, the number of possible chess games, the number of quantum states in the observable cosmos: these are all much smaller than a googolplex. So the leap from a googolplex to Graham’s number or TREE(3) is not something you’re supposed to feel in your gut. It’s something you track through the logic of how these numbers are constructed.

Where Large Numbers Actually Show Up

You might wonder whether any of these gargantuan numbers matter outside of pure mathematics. The answer depends on what you mean by “matter.” A googolplex itself does not correspond to any known physical quantity. No measurement in physics, chemistry, or astronomy yields a number anywhere near a googolplex. The largest physical estimates, like the number of possible arrangements of particles in the observable universe, top out around 10 to the power of 10 to the 122, which is in the googolplex neighborhood but still a precisely bounded quantity tied to cosmological models.

Graham’s number and TREE(3), on the other hand, arose as upper or lower bounds in combinatorial proofs. Graham’s number was an upper bound on a problem about the coloring of high-dimensional cubes, meaning the actual answer to the problem lies somewhere below Graham’s number but is otherwise unknown. TREE(3) bounds the length of certain tree sequences. These numbers were not invented to be impressive; they fell out of genuine mathematical arguments. The fact that they are so large reflects something real about the combinatorial structures involved.

The Busy Beaver function has a different kind of practical relevance. Because it grows faster than any computable function, pinning down its values for even modest inputs would settle open problems in mathematics. Knowing Σ(n) for large enough n would, in principle, resolve questions like the Goldbach conjecture or the Riemann hypothesis, because those conjectures can be rephrased as questions about whether certain small programs halt. This is not a realistic path to solving those problems, but it illustrates why the function occupies such a central place in the theory of computation.

The Difference Between Large and Infinite

Every number discussed so far, from a googolplex to Graham’s number to any value of the Busy Beaver function, is finite. Infinity is not a number in the same sense. It is a different kind of mathematical object entirely. You cannot reach infinity by adding, multiplying, or stacking exponents, no matter how many times you repeat the operation. The gap between any finite number and infinity is not a gap that can be bridged by making the finite number bigger.

That said, mathematicians do study different sizes of infinity, and the landscape there is just as surprising as the landscape of large finite numbers. The set of all counting numbers is infinite, but the set of all real numbers is a strictly larger infinity. Beyond that, mathematicians have proposed large cardinal axioms that assert the existence of infinite sets so vast they cannot be proven to exist from the standard axioms of set theory. These axioms extend the foundations of mathematics and serve as tools for pushing back against fundamental limits on what can be proven.

6PubMed Central. Large infinities and definable sets

Large cardinals are relevant to the question of “how big can numbers get” because they illustrate that even infinity itself is not a single thing. There is a whole hierarchy of infinities, each one provably larger than the last, and whether certain levels of that hierarchy exist depends on which axioms you accept. The study of large finite numbers and the study of large infinities turn out to be deeply connected: the same ordinal labels that index fast-growing hierarchies for finite numbers reappear in the classification of infinite sets.

Googology and the Culture of Big Numbers

There is an active community of enthusiasts, sometimes called googologists, dedicated to defining and comparing extremely large numbers. The name comes from “googol,” and the field (if you can call it that) sits at the intersection of recreational mathematics, proof theory, and competitive one-upmanship. Online wikis and forums catalog hundreds of named large numbers, each one defined by a notation system designed to outstrip the previous one.

The game works roughly like this: someone defines a new notation that allows you to describe numbers larger than any previous notation could express concisely. Others then analyze where those numbers sit in the fast-growing hierarchy, determining whether the new notation genuinely breaks new ground or simply repackages an existing level of growth. Serious contributions to googology often connect to real mathematics. The analysis of TREE(3), for instance, has drawn interest from both recreational number enthusiasts and professional logicians, because the function’s growth rate relates to deep questions in proof theory.

The culture around large numbers can seem whimsical, with names like “meameamealokkapoowa oompa” and elaborate back-and-forth about whose number is biggest. But beneath the playfulness is a genuine mathematical structure. Defining a truly novel large number requires understanding the hierarchy of growth rates, and demonstrating that your number exceeds someone else’s requires rigorous proof. It is one of the few areas of mathematics where the competitive spirit is worn openly and where amateurs regularly contribute ideas that professionals take seriously.

Can You Always Find Something Bigger?

The answer is yes, and the reasons are layered. Within ordinary arithmetic, you can always add one. Within the world of computable functions, fast-growing hierarchies can always be extended to a new level. And within the world of all possible functions, including uncomputable ones like the Busy Beaver function, there are always larger values waiting at higher inputs. Even the Busy Beaver function, fast as it is, can be outpaced by variants of itself: a “Super Busy Beaver” that uses a more powerful model of computation, for example, or a function that diagonalizes over all possible oracles.

At the level of infinity, the pattern continues. For any infinite cardinal, there is a provably larger one. And for any axiom system you choose to work in, there are statements about large sets that the system cannot prove, leaving open the possibility of extending the axioms further.

6PubMed Central. Large infinities and definable sets

A googolplex, then, is not just smaller than many other numbers. It is small relative to the full landscape of what mathematics can describe. It earned its fame not because it is anywhere near the top, but because it was the first enormous number most people ever heard of, arriving at a time when “a number so big you can’t write it down” was a novel idea. The mathematical world has long since moved on to territories that make a googolplex look like pocket change, and there is no sign that those territories have an edge.