25 horses, races of 5 at a time, no clock. Minimum races to find the top 3, and why exactly that many?

How to answer

7 races. Run 5 heats (races 1–5), then race the 5 heat-winners (race 6); that winner is overall #1. Only 5 horses can still place 2nd/3rd: the 2nd and 3rd finishers of the winners' race, the 2nd and 3rd from #1's heat, and the 2nd-place horse from the runner-up's heat — race those 5 once (race 7) to settle places 2 and 3.

Key idea: 5 heats + a winners' race fixes #1; one more race sorts the live contenders.