Smartphones aren't particularly well known for their action games, mainly because of the touch controls. The feedback just isn't there, and that problem mars quite a few classic ports. Even so, here's a few games that may tide you over if you feel that need to run and jump or do some action stuff on the iPhone. Note that there's a heck of a lot of games on the App Store and this list is nowhere near exhaustive. You probably already know all of them anyway. With this disclaimer, here goes:
Sequel to the classic smartphone game, Temple Run 2 is a simple dodging game with quite a bit of depth to it. Sure it's nothing but jumping, ducking and leaning one side or the other, but it does it well and it is satisfying. It's fast, furious and great fun, especially when you manage to hit a top score. The music is catchy and there's a ton of environments to run through. Try it, it's free.
The port of an extremely popular flash game on Armor Games' website, Kingdom Rush is a great tower defense game with a lot of content and depth, leveling system and so on. People love it for some reason, although there's been a few known problems with the compatibility on certain devices. Beware of that.
If you've had a smartphone in your hand, you've probably played Fruit Ninja. One of the most popular games on the App Store today, the game is known for being extremely satisfying to play due to the feedback given to you when you slice the fruit. If you haven't tried it yet, you owe it to yourself to. It's got incredibly great depth, considering that it's so easy to pick up and play.
If you get bored of the base version, there's the Puss in Boots version which everybody seems to love.
Making the transfer from simple towerbusting challenges on boring old 1g gravity, Angry Birds Space sees the particularly infuriated avians in space, at the will of gravity wells and the cold of space, having to destroy pigs on asteroids, other planetoids and other astral bodies. It's a clever twist and adds a lot of variety to the normal pull and release gameplay. Unfortunately, the game is rather short - although you pay 3 bucks for it,you only get the first two "worlds", after which you have to buy more levels with microtransactions. Why? Why not just sell it for 5 bucks with all the levels in it? Either way, it's a great game, especially to the Angry Birds fan.
Based on the popular 2010 original, Ridiculous Fishing was lovingly crafted over the course of two years by the guys of Vlambeer. It's got a very troubled history to its development, but you wouldn't see it if you only played the game - it's well-crafted, very fun, and one of the only games I know of where after you fish sea life out of the waters, you have to shoot it down with a minigun to actually catch it. There's a huge amount of unlockables, lots of levels, beautiful cubist art and a lot of fun in a very complete package.