Biggest numbers

There are about 10⁸⁰ atoms in the observable universe. This article is a countdown of real numbers — numbers used in actual mathematics, with actual definitions — that make "all the atoms in the universe" look like a rounding error. Not infinity: finite numbers. Each rung on this ladder is so far beyond the last that the previous champion becomes, in the next number's terms, indistinguishable from zero. And it ends with the strangest truth in all of mathematics: the biggest numbers aren't just unwritable — some of them can't even be described.^[1]

The ladder down — er, up

 googol      10¹⁰⁰          atoms in the universe: ~10⁸⁰  — already exceeded
 googolplex  10^(10¹⁰⁰)     can't be written in the universe — no room for the zeros
 Skewes'     e^(e^(e^79))   once the largest number in a real proof
 Graham's    g₆₄            needs a new notation — arrows — because exponents ran out
 TREE(3)     (finite)       makes Graham's number ≈ 0 — and grows *unfathomably* faster
 Busy Beaver BB(n)          uncomputable — outruns every function a computer can name
 Rayo's      Rayo(10¹⁰⁰)    biggest finite number ever defined in a math duel — by rule

8. Googol and googolplex — the warmup act

The googol (10¹⁰⁰, written ⠼⠂⠴⠘⠂⠴⠴10^{100}) was named in 1920 by Milton Sirotta — the nine-year-old nephew of mathematician Edward Kasner, who asked the boy what to call 1 followed by a hundred zeros. It's already bigger than the observable universe's atom count by a factor of a hundred billion trillion. The googolplex (10^{10^{100}}}) is a 1 followed by a googol zeros — you could not write it longhand in this universe even if every atom held a digit, and if you wrote it on paper it wouldn't fit in the observable cosmos at any physical scale. Google (the company) is famously a misspelling of the nephew's word — and their headquarters is the "Googleplex."^[2]

7. Skewes' number — the first monster in a real proof

In 1933, mathematician Stanley Skewes computed an upper bound on where ⠨⠏⠷⠰⠭⠾\pi(x) — the prime-counting function — first overtakes its approximation ⠠⠇⠊⠷⠰⠭⠾\mathrm{Li}(x): the bound came out to ⠰⠑⠘⠰⠑⠘⠰⠑⠘⠶⠔e^{e^{e^{79}}}, roughly 10^{10^{10^{34}}}. For decades this was the largest number to appear in an actual mathematical proof — the original "biggest number" champion. (Without assuming the Riemann hypothesis, his 1955 bound rose to an even larger tower: 10^{10^{10^{964}}}.) The honest postscript: mathematicians later proved the actual crossover happens near a "merely" astronomical number around 10^{316} — Skewes' bound was just the first time anyone proved there was one.^[3]

6. Graham's number — the number that broke arithmetic

⠰⠛⠰⠂⠀⠨⠅⠀⠼⠒⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠼⠒g_1 = 3\uparrow\uparrow\uparrow\uparrow 3

In 1971, Ronald Graham needed an upper bound for a problem in Ramsey theory — the mathematics of unavoidable patterns. The bound he published required a new notation invented for the occasion: Knuth's up-arrows, where each arrow is the previous operation iterated to insanity. One arrow is exponentiation. Two is repeated exponentiation (towers of exponents). Three arrows is repeated towers. Four arrows — the definition of g₁ alone — already produces a number whose digits can't be described by any tower of exponents writable in the universe. Then g₂ uses g₁ arrows. Graham's number is g₆₄ — sixty-four layers, each layer's arrow-count being the entire previous number. In 1977 Martin Gardner introduced it to the public as "a bound so vast that it defies comprehension," and Guinness listed it as the largest number ever used in a mathematical proof.^[4]

To feel one rung: ⠼⠒⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠼⠒3\uparrow\uparrow 3 is a power tower of threes — 3^{3^{3}}} — which is already 7.6 trillion. ⠼⠒⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠼⠒3\uparrow\uparrow\uparrow 3 is a tower of threes 7.6 trillion layers tall — a number too large for the universe to hold even its digit count. And ⠼⠒⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠫⠣⠒⠒⠕⠀⠼⠒3\uparrow\uparrow\uparrow\uparrow 3 — the humble g₁ — is a chain of those that itself defies iteration. Then come sixty-three more layers. Physicist John Baez's description is canonical: even the number of digits in the number of digits of Graham's number, repeated any conceivable number of times, is inexpressible.

And here's the beat that makes the rest of this list work: Graham's number is now routinely described as "effectively zero" compared to what's next.

5. TREE(3) — where comprehension ends

Kruskal's tree theorem (1960) proves that certain finite sequences of "trees" must eventually terminate — and Harvey Friedman turned that into the finite function TREE(n): the longest possible sequence of trees with n colors where no earlier tree contains a later one. TREE(1) = 1. TREE(2) = 3. TREE(3) is so large that Graham's entire 64-layer arrow construction is beneath notice — TREE(3) exceeds Graham's number by a factor expressible only with fast-growing-hierarchy notation that itself describes towers of towers of descriptions. It is finite. It is provably finite. And it is beyond anything the physical universe could express by any conceivable encoding. Friedman's related function SSCG(3) — "simple subcubic graphs" — exceeds TREE(3) by the same kind of incomprehensible margin.^[5]

The weird honest detail that makes TREE(3) beloved: the game is playable. You could sit down with paper and three colors of pen and start drawing legal tree sequences — you just could never finish. The universe ends first — not metaphorically: TREE(3) sequences outlast proton decay, black-hole evaporation, and every physical process that will ever occur anywhere. It's a game you can start and literally never, ever, end.

4. Busy Beaver — the number no computer can reach

The Busy Beaver function BB(n), defined by Tibor Radó in 1962, asks: among all n-state Turing machines that halt, which runs longest? BB(n) is that champion's step count. Here's the trap: BB(n) is uncomputable — provably. If any algorithm could compute BB(n) for every n, it could solve the halting problem, which Turing proved impossible. So BB(n) isn't just big — it grows faster than every computable function that exists or could ever exist. No formula, no arrow notation, no TREE() — nothing programmable can keep up. As of 2024, BB(5) was finally proved by a collaborative proof effort (47,176,870 steps); BB(6) remains uncomputed and is known to exceed 10↑↑15 — already beyond physical describability.^[6]

3. The paradox at the top — why "biggest" eats itself

The Berry paradox: consider "the smallest positive integer not definable in under sixty letters." That sentence defines it — in under sixty letters. Contradiction. This is why the biggest-number game can't just say "the biggest definable number" — that's self-eating. The real sport is definability within fixed rules: pick a language, count the symbols, name the largest number that language can express. Which is exactly what the final entry weaponized.^[7]

2. The Big Number Duel — MIT, 2007

On 26 January 2007, philosophers Agustín Rayo and Adam Elga faced off at MIT in the "Big Number Duel" — write the biggest finite number you can define, and the rules forbid "Elga's number plus one." Elga opened strong. Rayo answered with the most audacious definition in the history of the sport: "the smallest number bigger than any finite number named by an expression in the language of first-order set theory with a googol symbols or less." Call it Rayo's number — ⠠⠗⠁⠽⠕⠷⠼⠂⠴⠘⠂⠴⠴⠾\mathrm{Rayo}(10^{100}). Every number in this article — Graham's, TREE(3), SSCG(3), Busy Beaver values up to any describable bound — can be defined in far less than a googol symbols of set theory. Rayo's number is bigger than all of them — by construction. It won the duel.^[8]

1. There is no winner — and that's the punchline

The honest crown belongs to a fact, not a number: there is no biggest number — add one and you're bigger. What this countdown actually ranks is humanity's ability to describe largeness. The ladder runs: notation (googolplex) → iterated notation (Graham's arrows) → provable termination (TREE) → the uncomputable (Busy Beaver) → definability itself (Rayo). Each rung exists because mathematics kept needing numbers bigger than its notation could hold — and solved the problem by inventing more notation. The biggest numbers aren't records. They're a history of the human imagination discovering it had, once again, run out of room.^[9]

The ledger

Frequently asked

• What is the biggest number in the world? There isn't one — add one. The biggest named number is generally taken to be Rayo's number (2007).
• Is TREE(3) bigger than Graham's number? So much bigger that Graham's number is effectively zero by comparison — TREE(3) uses a faster growth regime entirely.
• What's bigger, googolplex or Graham's number? Graham's number — beyond any notation a googolplex could help express.
• Can Busy Beaver ever be computed? Provably never, in general — computing BB(n) for all n would solve the halting problem.

See also

• Scientific notation
• Zeptosecond
• Zeptoverse
• Halting problem