To OP:

Personally, this behavior would bug the heck out of me.
Unfortunately, I agree in part with what hvert says. The reason being, you're only going to stress yourself out by acting this way and getting mad at him for his behavior. The bad news is that it's most likely not going to change him.

If it were me, I would probably have a conversation about this with him ONCE. And say everything I want to say then and there. Get it all out, and don't repeat it later.
I would probably say something along the lines of, If you don't want to get your *** in jail, you'll take care of this. I'm sure you don't feel it's a big deal, but the cop that pulls you over will disagree with your opinion, and I can guarantee you your opinion will NOT trump his.

This stuff does upset me, but you know what? One day it'll catch up to him. If he does get pulled over with an expired license, he WILL be taken to jail asap - especially if it's been well over a month and the cop will see he's had all this time to get this taken care of but he hasn't.
The time that happens, he'll remember the one conversation you've had.

It's not your responsibility to act like his mom and do things for him. He's a grown man and he needs to learn responsibility, and not through you.

If he does end up in jail and it ends up inconveniencing you, like say, you have to go bail him out or something, then I would think it's time for another serious talk. One of those, You need to realize that your behavior is affecting the people around you and that's not fair. Put your foot down and let him know you won't be paying for his shortcomings. You're both adults.

From then, it's all up to what kind of attitude he takes with you. If he apologizes and realizes his behavior is bad, then there's hope that he's maturing. If he gives you poor attitude about this, and acts like he's got all the right answers - that's when I'd advise you start considering ending the relationship.

As it is, there wasn't enough information on your post about his attitude, which I feel is the key point here.