What Is the Largest Number You Can Name?

In January 2007, a crowd gathered at MIT for an unusual event: the Big Number Duel. Two philosophers, Agustín Rayo of MIT and Adam Elga of Princeton, took turns writing numbers on a board. Each new entry had to be larger than the last, and each had to be clearly defined.

It started with long strings of 1s and quickly escalated to exotic notation. Rayo won with a number defined using the language of set theory itself, now called Rayo's number.

"Name the biggest number" sounds like a children's game. Played seriously, it leads to some of the deepest ideas in mathematics and computer science.

The Winner Isn't Who Writes the Most Digits, but Who Uses the Most Powerful Idea

You can always add 1 to any number, so there's no largest number. But the game has a rule: you must name a number in a limited amount of space, say 15 seconds or one index card.

In that game, writing 9999999… loses badly. What wins is choosing a more powerful operation. Each great leap in big numbers came from a new way of building on previous operations. Computer scientist Scott Aaronson explored this in his 1999 essay "Who Can Name the Bigger Number?"

Level 1: Exponents

Multiplication is repeated addition. Exponentiation is repeated multiplication:

10¹⁰⁰ = a googol

A googol has 101 digits. It's already far larger than the estimated number of atoms in the observable universe, around 10⁸⁰. A googolplex is 10^googol, a 1 followed by a googol zeros. See How Many Digits Does a Number Have?

Level 2: Towers (Tetration)

Repeated exponentiation is called tetration. In Knuth's up-arrow notation, introduced by Donald Knuth in 1976:

3↑3  = 3³ = 27
3↑↑3 = 3^(3^3) = 3²⁷ = 7,625,597,484,987
3↑↑4 = 3^(3^(3^3)) = 3^7,625,597,484,987

That last number has about 3.6 trillion digits. Add a third arrow:

3↑↑↑3 = 3↑↑(3↑↑3) = a tower of 3s that is 7,625,597,484,987 levels tall

Each arrow iterates the operation before it. Compute 3²⁷ with the scientific calculator.

Level 3: Graham's Number

In the 1970s, Ronald Graham and Bruce Rothschild worked on a problem in Ramsey theory about coloring connections between corners of high-dimensional cubes. Graham's number emerged as an upper bound in related work, and Martin Gardner publicized it in 1977:

g₁ = 3↑↑↑↑3
g₂ = 3↑↑…↑↑3   (with g₁ arrows)
…
Graham's number = g₆₄

At each step, the number of arrows is the previous enormous number. There isn't enough room in the observable universe to write its digits, or even the number of its digits. Yet we can compute its last digits: …2464195387.

Level 4: Faster-Growing Functions

Mathematicians compare big numbers by the growth rate of the functions that produce them. The Ackermann function, defined in the 1920s, grows faster than any function built from ordinary loops of fixed depth. TREE(3), from a combinatorial game with labeled trees, dwarfs Graham's number so completely that Graham's number is effectively zero by comparison.

Level 5: The Busy Beaver

Here's where computation enters. In 1962, mathematician Tibor Radó defined the Busy Beaver function.

Consider every Turing machine, the simplest model of a computer, with n internal states. Some run forever. Among those that eventually halt, which runs the longest? That number of steps is BB(n).

An Insider Reference: BB(5) Is Settled

Determining BB(5) required proving that every one of the many five-state machines either halts within 47,176,870 steps or never halts. In 2024, the online collaborative project bbchallenge, involving many contributors, announced it had proved BB(5) = 47,176,870, with the proof checked in the Coq proof assistant.

BB(6) is already known to be larger than a tower of 10s more than ten levels high, and Aaronson and others have argued that some Busy Beaver values will likely never be determined.

Why? The Busy Beaver function grows faster than any computable function. If you could compute it, you could solve Turing's halting problem, which is impossible. It's the point where "naming a number" meets the limits of computation.

Where Naming Breaks Down: Berry's Paradox

Consider:

"The smallest positive integer not definable in under sixty letters."

That phrase defines a number, in under sixty letters. Contradiction. This is Berry's paradox, published by Bertrand Russell in 1908 and credited to Oxford librarian G. G. Berry.

It shows that "definable" must be used very carefully. Rayo's number avoids the paradox by referring to definability in a precise formal language, a strategy made rigorous by logicians.

Two Concepts Worth Knowing

Hyperoperation

Hyperoperations build a ladder of operations: addition, multiplication, exponentiation, tetration and beyond, each repeating the previous one. Up-arrow notation climbs this ladder.

Uncomputable Function

An uncomputable function has values that no algorithm can calculate in general. The Busy Beaver function is the classic example, and it outgrows every computable function.

Quick Answer: What Is the Largest Number?

There's no largest number, since you can always add 1. But in "name the biggest number" contests, huge numbers come from powerful operations: a googol (10¹⁰⁰), Knuth's up-arrows, Graham's number, TREE(3), and the Busy Beaver function, which grows faster than any computable function. BB(5) = 47,176,870 was proved in 2024.

Try Them Yourself

• Scientific Calculator: compute 3↑↑3 = 3²⁷
• Numbers to Words: name 7,625,597,484,987 in words
• Logarithm Calculator: count digits of 3↑↑4
• Base-10 Logarithm Table: orders of magnitude
• How Much Is a Trillion?: start with everyday big numbers
• The Weirdest Numbers in Mathematics: more strange and enormous numbers

Challenge a friend: each of you gets 15 seconds and one index card to write the biggest number you can clearly define. Afterward, compare which ideas, not which digits, won.