In Who can name the bigger number?, Scott Aaronson gives two examples of fast-growing functions:
- The Ackermann sequence, defined specifically as
A(1)=1+1, A(2)=2*2, A(3)=3^3, etc - The family of Busy Beaver functions.
Wikipedia also has examples of fast-growing functions, like the TREE sequence and the SCG function. All of these are discrete functions. Even "simpler" fast-growing functions, like $2 \uparrow^n 2$, are discrete. We could define continuous functions via interpolation but that feels unnatural.
Are there any very fast growing functions, faster at least than BB, that are also "naturally" continuous?