Since I can only put one pokemon in a gym, I feel like there is no point to training a gym beyond enough to stuff my pokemon in it. At the same time, I feel there is probably more strategy to it than I am seeing on the surface. Is there any reason that I should be interested in increasing a gyms level beyond current + 1?
Well the higher the prestige on the more time and work it will take another team to take it down. As well if you raise it up it will be closer for another player of your team to add a Pokemon of theirs also making it harder to take down.
Both of those mean your Pokemon will stay there longer and you can continue getting rewards from that gym. Plus there's the small bonus of the more you raise the prestige you get some experience from it too but it's not a whole lot.