The ITTM Busy beaver is to the Busy beaver as the Infinite-order Xi function is to the Xi function.
We define a combinatory language with countably infinite symbols such that there is a bijection between these symbols and the natural numbers (including 0).
0x := \(\iota\) x as in Chris Barker's iota combinator. Nxyzw := Where N is a natural number denoting a symbol, \(\Omega\)xyz (as in the oracle combinator) if x is well founded, w otherwise.
Now that we have our language, let's just reuse the definition for the Xi function.
If we start with a string of n symbols and we beta-reduce it, the largest possible finite output is called \(\Xi_{\infty}\)(n)