A story I read on the Internet: Someone (let's call him Michael) played a Samurai (let's call him Bob) who was supposed to defend a particular girl. Blah blah blah, girl gets kidnapped into some tunnels. Now, a normal party would take the time to search for traps and disable them, and that's what Michael normally would do as well. But Bob would place his life below his duty, and would willingly sacrifice his life to ensure the safety of his charge. Instead of having the party rogue disable the traps, Bob just charged straight through them, saving his party valuable time, saving the girl, etc etc. Along the same vein, Bob would be willing to sacrifice his life to save the rest of the party, and at one point I believe he did.
Note that this works because Michael wasn't upset about Bob dying, and because Michael didn't just have Bob constantly Charge&Targ-ing and messing with the fun of others.
A generalized example: What point does your character deem too risky? If a wizard, is he/she stingy with spells or does he/she blast through them with great At what point (be it HP left, or status effects or number of spells left) does your character say "Oh god, I'm bookin it!"? What will your character go beyond that point for? So on and so forth. Combat has plenty of Roleplaying possibilities, not just Rollplaying possibilities.
A personal story: Two years ago I made a Warforged Paladin for a 4th Edition game.[footnote]Fourth edition sucks, but it's the party that matters. I've had more fun with that group than I've had with any other game because the people involved were very fun to be around. Well, except the DM but he didn't ruin TOO much of the fun.[/footnote] I determened at character creation that he (Thraisus, by the by) was
literally created yesterday. When our party met with the steriotypical nubile female elf questgiver in danger, Thraisus responded with "May we provide you with assistance, Sir?".[footnote]Note that it took about 5 out of game weeks for another character to explain the birds and the bees to Thrasius. The next week he pleasantly asked a fat man when she was due.
Thrasius was sad and confused when the man started yelling at him.

[/footnote] And when we got to the steriotypical dungeon, the rest of our party spent about ten real life minutes trying to figure out what to do. It seemed obvious to me that there would be some sort of trap along the way, but the kids weren't even addressing that. Besides, Thrasius wouldn't know that (still 1 day old at this point, remember?), so at that point, I looked at the DM and said "so Thraisius would be walking forward." as an aside to the group "Hey, is anyone gonna stop me?" I move him five feet forward. "Hey, is anyone gonna stop me?" I move another five feet. "Guys? Hello?" I move another five feet.
*Boom* Pittrap. *Boom* Spiders everywhere*
And much terror was had by all.