I cannot answer your question about the UK.
For the Schengen Zone the rule is that US citizens are limited to 90 days (not 3 months!) within any 180-day period. Every day you must look back 180 days and ask yourself, "Have I been in Schengen countries for 90 days?" If the answer is "Yes", you must leave that day. Both the arrival day and departure day count. By taking a long break (or multiple breaks), it seems possible to spend 180 days each year in Schengen countries. What I don't know is whether questions might be raised about repeated long visits. Might the Immigration authorities suspect under-the-table employment, especially if the visitor was of working age?
I take annual 88- or 89-day spring/summer trips to the Schengen Zone but have not made two such trips in the same calendar year.