Completing the Friend of the World aspiration. When your sims are just as asocial and busy with other stuff as you are in real life, keeping so many friendships at once becomes quite the challenge. Next up on my list is Serial Romantic, which is just as annoying, but at least not as hard because you don't need to be in all those relationships at the same time.
If we're talking about actual achievements that come with notifications and all that, I recently got one for having fifteen broken objects on the lot. Not that it was particularly hard (the Gremlins lot trait does that to you), but I had no idea this was a thing, and it came as a rather big surprise.