My friend and I have discussed this at length. What if you asked a Meeseeks to complete a task which is physically impossible, rather than just hard? "Mr. Meeseeks, can you please discover what lies at the edge of the Milky Way?"
Obviously Meeseeks try their very hardest to complete their objective, but I believe that they employ two strategies if their goal seems unreachable. This is all speculation, as aside from the wiki entry already linked, we don't have much information about them.
- Mr. Meeseeks will try to create more of themselves to complete the objective.
This ensures that even if the original Meeseeks degrades beyond the point of assisting the summoner, there is still a way to complete the objective. This is employed when the goal still seems achievable.
- Mr. Meeseeks will attempt to eliminate the source of the request
I believe that were you to ask the above question, Mr. Meeseeks would try to kill you immediately. There is no conceivable way to fulfil the request, so the only real course of action if to fulfil the request by making it no longer valid. Otherwise the univerise would be litered with dismembered and decaying Meeseeks bodies.
Mr. Meeseeks will always attempt one of the above before decaying to the point of inaction