Never really had a close call with a deer on the bike. (I did have one last weekend in the car, though.) Back when I lived in Indiana I was at a race and a guy had a run-in with a deer. I don't think the deer had any injuries. The rider and his bike got the worst of that situation.
I'm more worried about squirrels. They may be small, but they're not afraid to get close to you and it's hard to tell what the hell they're doing. The animal itself doesn't have to do the damage- the road is plenty capable of that.
Wild horses are also a concern around here, but they tend to be slow-moving and predictable.