The thing about these unorthodox players is that everyone assumes that they are LAGtards just hurling chips around. This isn't necessarily the case. Remember when Patrik Antonius utilized an unorthodox style during the WSOP
a few years back? He seemed to be raising pre-flop with anything. When your quality hand missed the flop and he fired a c-bet, there was no way to know if he hit because it was just about impossible to put him on a hand. He could have anything.
What makes these guys good is knowing when to lay a hand down and knowing when to keep putting the pressure on. If they see that you've been playing tight, then they know a you probably missed the flop when a bunch of small cards hit. When you call on the flop, they know it's likely that you're hoping to hit an overcard. When one doesn't come, they keep firing. And with each street, as the pot increases, so does the size of their bets. It becomes harder and harder to make those calls, counting on hitting one of those overs, knowing that this guys range is wide enough to have hit any card on the board already.
I think the key to playing against these guys is to open your range up slightly. You don't need try to outLAG them, especially if you're not comfortable playing that kind of style. And when I say open your range up slightly, I mean against this particular player. I'm not saying you should be playing bingo against half the table. If someone else wants to mix it up with this guy, get out of the way and let him do it. Just because you're including more hands in your range doesn't mean that you have to play them every time you get them or from every position.
One thing I do against these players is to defend my blinds a little more strenuously than I might otherwise. Who says their random two cards are gonna be better than my random two cards. Let's play. I will definitely play back at these players pre-flop when I'm dealt hands in my normal raising range. Normally I wouldn't play hands that I wouldn't be willing to raise with. These hands are now good for a 3-bet against this particular player.
And switch it up. Don't become predictable. For the most part you want to be playing fit or fold. If you don't flop it, drop it, unless you pick up a good draw like an open ender or a flush draw. If you hit and are first to act, try check raising. See if that tames his aggression any. Once it does, start leading out when you hit. Try to make him pay if you can, but don't let him see any cards for free. Change it up from time to keep him off balance. Before long, he'll be the one playing tight.