Five infinite numbers

Each page starts on the end of a formula for a huge number and zooms out for ever, faster and faster, until you click — then the formula gets its outermost level and you are looking at one whole, finished number. Left to right, each way of writing it is a whole tier more powerful than the one before.

The rungs count steps and folds. A step repeats the operation before it (+, ×, ^, ↑↑, …); a fold, ω, jumps to “as many steps as the number itself”; ω² folds the folds; ε₀ is where ω, ω^ω, ω^{ω^ω}, … is heading, and Γ₀ is where subscripts inside subscripts inside subscripts are heading. N is how many levels deep your finished number is — you choose it by when you click.

Towers

towers of 10s, stacked with braces

rung N

past Graham’s first step, g₁

Arrows

Knuth’s arrows: Graham’s diagram, nested

rung ω + N

beats Graham’s number, whenever you click

Chains

Conway’s chained arrows, nested

rung ω² + N

beats any Conway chain anyone could write out

Brackets

brackets that fold whatever is inside them

rung ω^{ω^{⋰^ω}} (N high)

climbs toward ε₀ — past everything the other three can reach

Subscripts

brackets whose subscripts say what kind of fold they are (Veblen’s hierarchy)

rung φ_{φ_⋱} (N deep)

climbs toward Γ₀ — past everything the brackets reach, whenever you click