[kamkha]

Close! You swapped an operation there: it'd be (2^8 button possibilities) ^ (5 minutes * 60 FPS), or about 10^43348.

A bit of a bigger space to search ;)

[mrob]

Only 2^7 button possibilities on a real controller, because you can't press up and down, or left and right, at the same time.

[LocalH]

Worn controllers can allow it to happen, however, so I don't think it's worth discounting.