Limited Advancement Opportunities: It's pointed out several times that he is permanently stuck in his position as associate due to his complete refusal to play by corporate rules, but he's simply too good and valuable as a trump card to fire him.
Nice Hat: Not necessarily always a hat, but Alan is particularly fond of wearing strange things on his head. Ornaments (that light up, no less), mistletoe, wigs, coonskin caps...
The Only One: He's the go-to man when it comes to impossible cases, since his originality may be the thing to save the day. Also, a lot of such cases are so fucked that there's nothing he can do to screw it up even more.
The Not-Love Interest: For Alan, to the degree that when Denny decides he wants Alan to inherit his estate when he dies, they get married to seal the deal.
Showy Invincible Hero: Denny never lost a case and he never does over the course of the show, but his insane behavior in court only makes it more fun.
Straw Character: Denny Crane is a strawman political caricature of a conservative gun nut, although this did allow him to save the day a few times. Keeping guns in your office seems so much less crazy when you just used them to shoot the man about to kill your friend.