I'm trying to grow mint, and lettuce in the window sill, and was wondering if just epsom salt with water is enough to keep the "clippings" alive for giving a crop.
If you have good soil, possibly it is enough for a year. According wikipedia epsom salt (Magnesium sulfate) is MgSO4·7H2O so it will give only few substances to the plants. On the wikipedia article is also described that epsom salt could be used as fertiliser in case of deficit in magnesium or sulfur.
But the epsom salt lacks a lot of other important nutrients: see Plant nutrition (Wikipedia). The most important components are N, P and K, but also other minerals are essential (but on very small doses).
I expect lettuce to have high requirements on fertilisers. Mints growths on poor soils, so it has less requirements, but for the flavours, it still needs minerals.