My favorite types of pre-iPhone flash games were always building puzzles, a genre that is just minimally represented on the App Store. Tiki Towers [$1.99 / Free] was the first one I played and beat, then Moonlights [99¢] came along offering similar gameplay. While not a bridge building game, Geared [99¢ / Free] tickles the same part of my brain as these other iPhone favorites of mine.
In Geared, players are tasked with making the virtual gearboxes work across 80 included levels. Each challenge has a spinning gear and a stationary gear. Utilizing the five different gear sizes, each available in limited quantities, you need to arrange the gears to power the original stationary gear. The big benefit that Geared has over other similar games is that gears can be placed anywhere in game. There is no grid that they snap to or any other alignment requirements other than making sure that the teeth of the gears line up with each other to spin.
Like most puzzle games with static levels, the first few you come across introduce you to the gameplay elements and are extremely easy. It doesn't take long to eventually get to puzzles that will have you moving gears all over the place trying to find a gear placement that works. Eventually areas that can't have gears placed in them appear in levels, and on some of these challenges with these zones require you to utilize the gravity in the game to drop gears through them to wind up in the right spot.
The one main complaint people seem to have about the game is that some of the advanced levels require pixel-perfect accuracy in placing gears. Depending on how neurotic you get about construction games you could either see this as a good or bad. Personally, I'm appreciating the challenge offered in the game, but I can certainly understand the frustration some are experiencing.
If you like these kind of games, you owe it to yourself to at least download the lite. If you're new to the genre, I'd recommend getting started with both Geared Free and Tiki Towers Lite