Ah! My internet connection died while I was writing my answer, and it seems Mark beat me to the punch. So allow me to elaborate on the horseshoe orbit of Epimetheus and Janus, if I may.
As previously mentioned: three planets in such close proximity just isn't plausible. There are several realistic alternatives for 2 planets however, including a binary planets and the co-orbital horseshoe.
Binary planets are more in the vein of what you're looking for and relatively easy to explain. A binary/double planet is a system with two planet-sized objects, orbiting a common center of shared mass, orbiting a star. But we don't know a lot about systems like this.
(For more, see: https://en.wikipedia.org/wiki/Double_planet)
But I think the other option - the horsehoe orbit - is certainly notable and just as, if not more, romantic. We also have a very good example right in our very own solar system.
Epimetheus and Janus, both moons of Saturn, are co-orbital. Technically. It’s weird.
Their semi-major axes (think average distance from Saturn) are slightly different. Not a lot - less than either’s diameter - but still a bit.
Kepler’s laws of planetary motion dictate that the moon that is slightly closer to the sun will complete its orbit slightly faster than the other. In this case, the inner moon completes its orbit 30 seconds faster than its partner. That might not seem like a lot, but every day, the distance grows… and it adds up.
Eventually, they meet each other. And when they meet, it gets even weirder.
They switch orbits.
As they tug on each other, the inner moon loses speed, falling into a slightly more distant orbit, while the outer moon accelerates into a closer orbit. The cycle repeats.
It looks something like this:
Imagine seeing this from the surface of either moon: standing on Epimetheus, watching as Janus races toward you, getting closer and closer, larger and larger, then fall away into the darkness. Or maybe from the surface of Saturn? It would look weird.
There’s a lot to play with here. Who is to say they have to be moons? It could be possible for two planets to behave this way.
The idea of a binary planet + lone planet together in a horseshoe orbit springs to mind. I really doubt it's plausible, but it sounds neat. And who knows? maybe there's a way to make it work somehow. There's a lot of strange stuff out there.
You can find more information about horseshoe orbits here:
https://en.wikipedia.org/wiki/Co-orbital_configuration#Co-orbital_moons (this is the link Mark provided)