Similar idea, but with Turing machines rather than physical computers.
Simplified a bit, the n-th busy beaver number is the number of 1s which can be written out by an n-state Turing machine.
For 2-symbols and n-states, the sequence goes: 6, 21, 107, >= 47176870, > 7.4×10^36534, > 10^10^10^10^18705353 — and those inequalities are because the numbers themselves cannot be computed directly, you have to run the program to see what it does.