User blog comment:LittlePeng9/ITTM galore/@comment-10429372-20140318165300/@comment-1605058-20140318185719

I don't think multicolor machines have been investigated ever (after reading through 6 papers on the topic I haven't seen any mention of it), but taking limsup is most straightforward method of extending this while keeping 2-color transitions consistent.

I don't think there is a way of translating Busy Beavers to Timeless Tigers which would give us any decent bounds. I can think of a way to translate TM making \(n\) steps to ITTM making \(\omega n\) steps, but this is meaningless, as already 1-state ITTM can go above it.