Right now I have an Intel i3-3220 and a GTX 660. My motherboard can support up to four GPUs for a four-way SLI.
Would I need a more powerful CPU if I ever wanted to add one more GTX 660? What would be the cheapest Intel CPU that could handle two GTX 660s?
And if I need to upgrade my CPU along with a new GPU anyway, then would it make more sense to completely replace my current GTX 660 when I want to upgrade, and get a better GPU and CPU, or just get a new CPU and another GTX 660, and SLI the two (or three, or four) cards?
You don't need to upgrade your CPU along with a graphics card addition. You'll still receive maximum benefit of the GPU power, but what you may find is that your CPU is now your bottleneck. Probably true in this case- but it depends on what you're looking to do with your i3-3220, its fairly recent and I don't expect that it will cause you any problems. It'll likely become your bottleneck with 2x 660s, but thats fine... you're always going to have a bottleneck and you can only plug one hole at a time.
ie. you could get the latest/greatest CPU and then your bottleneck is right back to graphics again... then you could buy 2 more 660s for 4x SLI, and then kick the bottleneck right back to your CPU... unreasonable example- but still true, even the guys with $3000 PC's have bottlenecks.
Is there a way for me to determine roughly whether one CPU is going to bottleneck a GPU, or multiple GPUs in my case?
Not really- it's too variable and dependent on what you are doing. Honestly, it is far better to be CPU restricted than GPU restricted.
You have a recent gen Intel chip, you're good man. I can't see any realistic situation where you would be kicking yourself for having an i3-3220.
For example, I have an i5 - 750 Lynnfield CPU (About 3 generations behind now) running with a GTX 770 GPU. The CPU is an obvious bottleneck in most situations, but it doesn't matter- the bar is still set pretty high between the two components. There just isn't a way to justify a big expenditure for a 10-15% overall performance increase. Your i3-3220 probably performs pretty similarly at stock, as my i5 750 does overclocked. You'll be fine.