I know it's very vague, but as I'm only a novice Googologist, I just wanted to know if the general idea is plausible.
Step 1: Create a fast-growing function that outputs not finite numbers, but ordinals
Step 2: Use this function to define a very large ordinal
Step 3: Create a fast-growing function using the fundamental sequence of that ordinal
Step 4: Use this new function to define a very large finite number