This guy is in Medical Pavilion. No mines. No gel. No trap bolts. No Winter blast. Available plasmids: Electroshock, Incinerate, and Telekinesis. He likely has Telekinesis and Electroshock equipped, and doesn't have access to a Gene Bank.
Here's what I did:
1) If you have any friendly turrets, try to fight near them.
2) If you have telekinesis equipped, grab an exploding canister or barrel. Your first attack in the fight should be to launch this at the big daddy.
If you want to use this type of attack, it must be your opening move, since it's a bitch to launch these at a big daddy who is already interested in killing you.
3) Electric buck, is your preferred ammo type. Don't let the big daddy get near you unless you are prepared to stun him.
4) You can stun him with electroshock, run up to him, shoot him twice with the shotgun, and run for cover. Repeat until you kill him. I'll admit that this isn't very inventive, but this Big Daddy fight isn't like the rest. Once you're in Neptune's Bounty (the next area) you get more plasmids, more weapons, and your options really open up. This will probably be your only toe-to-toe Big Daddy fight in the game. Also, the Big Daddys get tougher in every area.
Beyond Medical pavilion, any of the suggestions posted above become plausable. My favorites:
1) Use Hypnotize Big Daddy on a big daddy. He'll follow you for a while. Take a single shot at another big daddy. Run away. When you come back you will see one of 2 things:
a) Two dead big daddies. Rarely happens with Rosies, and never with Bouncers. Usually results from the "winner" stepping on the loser's proximity mine.
b) One big daddy wins, but has a sliver of health. This is the most likely outcome, since all big daddies in a given area are of the same type, and toughness. So the matches are almost always even, unless one big daddy has a terrain advantage. Reward the winner (whether it is hostile or friendly) with a single shot of electric buck.
Note that these fights take a while, so the hypnosis will likely wear off before the fight ends. The color of the port hole in the big daddy's helmet will be green for friendly, yellow for neutral, or red for hostile. This is my favorite method of killing big daddies. For the cost of an eve hypo, and 2 bullets, I get 2 dead big daddies. Most methods that rely on proximity mines, grenades, and other manner of pricey munitions makes the cash that big daddies drop barely cover the cost of bringing them down. And you only have one dead big daddy to show for your efforts. I often use this method for killing big daddies even after all little sisters in an area are rescued, since it's a big money maker.
2) Use telekinesis to grab an exploding barrel, or gas canister. Carry it to an appropriate area, and gently set it down (press "v"). Stick about 3-5 proximity mines on it (they are sticky). Use telekinesis to pick it back up, and launch it at the big daddy. This will one-shot big daddies at lower levels, and seriously hurt them later in the game.