Rayo(Graham's Number passed to the busy beaver function, with the result of that computation passed to the busy beaver function - repeat a number of times equal to Graham's Number)

Apply TREE to 9, BIG-G times. Where TREE is Friedman's function in the context of Kruskal's tree theorem, and BIG-G is TREE applied to Graham's Number.

The biggest number proposed by another answerer plus one

Rayo(Graham's Number passed to the busy beaver function, with the result of that computation passed to the busy beaver function - repeat a number of times equal to Graham's Number),  then square the answer. 

The smallest integer that does not have a successor

The smallest integer that does not have a successor plus 9999999/10000000

Apply TREE to 9, BIG-G times. Where TREE is Friedman's function in the context of Kruskal's tree theorem, and BIG-G is TREE applied to Graham's Number... then square the answer. 

BB=Busy Beaver Function. Let BBC(N) be BB composed with itself N times onto N, that is BB(BB(... BB(N)...)) with N layers of BB. Similarly let BBCC be BBC composed with itself N times onto N. Answer is BBCC(BBCC(BBCC(BBCC(Rayo's number))))

The market did not say “largest of the answers”, it said “largest number.” There is none.

notation F[n] = composition of func F with itself n times. R=RayosFunc

Extending Graham's-number-like towers
Lyr(1)=R(9!), Lyr(n)=R[Lyr(n-1)](9!)
Twr(1)=Lyr(9), Twr(n)=Lyr[Twr(n-1)](9)
Row(1)=Twr(9), Row(n)=Twr[Row(n-1)](9)

Row[Row(9)](9)

The biggest number proposed by another answer plus two.

This market resolves to the answer that describes the largest number.

Manifold automatically limits answers to 240 characters, so your definition must fit in that length. If this limit is changed in the future, I'll stick to 240; anything longer will be invalid. Answers that attempt to circumvent this length limit by providing a reference to some other location are invalid.

Answers that do not uniquely and unambiguously define a single real number are invalid.

Answers that use mathematical symbols or technical terms are only valid if I can easily find a clear definition of that symbol or term. If I have to ask you for clarification in order to know what number you're describing, that answer is invalid.

If, when this market closes, I am unable to determine which answer is the largest, I will resolve to multiple answers in proportion to the probability I assign that that answer is in fact the largest.