We simply say "four times," "five times," and so on. And actually, in America, you won't hear "thrice" very often. It is very formal. Most people will just say, "three times."

Nothing! These three are the only words of their type, and no further terms in the series have ever existed...

In Ireland you won't hear anyone use 'thrice' just like in America, to be honest I don't think anyone anywhere uses it anymore unless someone is trying to sound posh. It goes: Once,twice,three times,four times etc.