Most of my projects are built around interfacing microcontrollers with I2C sensors. I've never had too many problems getting I2C communications working. I think a lot of it is psychological. Without a logic analyzer you can feel a bit helpless when you're not succeeding in connecting to an I2C peripheral. However, there's only a few things that plausibly can be going wrong with something as simple as I2C. (You have the wrong address, you have it wired up wrong, or you're misusing the uc's I2C peripheral.) It's quite feasible just to work through that list of possibilities until you find what's wrong. It does require a certain amount of faith, as you generally see no indication of progress at all until you get it working.

I'm not denying at all that a logic analyzer can be helpful. I'd just encourage people who can't justify the expense to have a try without one.

Edit: That said, I see that low-end logic analyzers are actually pretty cheap. I should probably get one!

Word of warning. I lost several days of debugging what looked like a working serial bus using a logic analyzer. I was about to give up when a friend asked what it looked like under the scope. The problem jumped right out at me. Certain bit patterns would distort the signal and I wasn’t crossing a threshold. Solution was trivial to fix after that, just replace a pair of pull up resistors and the scope showed good squares again. Had a similar problem with an optocoupler after that. Logic analyzer hid the fact that the optocoupler was too slow, and none of the leading edges were ever square. Ended up replacing that outdated part with a similar priced newer part. My client at the time was strggling with a whole batch of these optos for a number of SKUs that were within spec, but the variance was so high that it was common to get a batch that weren’t good enough for his designs.