Prince a Hermit for unlimited Madmen (you don't trash the Hermit)
You also don't discard the Hermit from play because it's set aside with Prince, and stays there. So you never trigger the effect, "When you discard this from play... gain a Madman." Sorry.
No, you do discard it from play. That triggers two effects: Prince's and Hermit's, and you get to decide the order. If you choose wrong, you'll lose the Hermit, but if you don't, you get to keep it.
To expand on this a bit more (this post will be worth $3 more than Awaclus'), this is what happens:
1. You discard Hermit from play (without buying anything). This triggers Prince and Hermit.
2. You decide which to do first. If you choose Prince (which is all I'll worry about here):
3. You set the Hermit aside, being able to play it again next turn.
4. You try to trash the Hermit, but can't because it has lost track of itself. You still gain the Madman.
PPE: Yes, a card can lose track of itself. Most cases of losing track are a card losing track of itself (TR-Mining Village for instance).