Limited Advancement Opportunities: It's pointed out several times that he is permanently stuck in his position as an associate at Crane, Poole & Schmidt 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 best friend.