Are there any time-periodic solutions to Einstein's equations apart from black holes? Are there any solutions to Einstein's equations which are periodic in time?
A black hole only has mass, charge and angular momentum according to the no-hair theorem. (Although this might just mean in the limit as time goes to infinity because when two black holes collide the shape of their combined horizon would be extra information).
Would a binary black hole system be periodic? I'm not sure because it might give off gravitational waves and collapse. (But there is also the time reverse of this where a single black hole absorbs gravitational waves and splits in two!)
What about a pair of black holes in a Universe that is expanding so fast that they always stay the same distance apart?
 A: In the paper Non-Existence of Time-Periodic Vacuum Spacetimes it is said that,

This paper addresses the question whether there exist asymptotically flat solutions of the Einstein vacuum equations which are “periodic in time” in a suitable
  sense. We show that any such solution must necessarily be stationary near infinity.
  Thus, genuinely “time periodic” solutions do not exist, at least in a neighborhood
  of (null) infinity.

Of course, in the general case, one can write down any time periodic metric, and compute the stress-energy tensor, thus offering a "solution" for that particular stress-energy, if physically sensible.
In the book, Exact Solutions to Einstein's Field Equations, there are plenty of solutions with a $\partial_t$ Killing vector which are thus technically periodic in time.
A: Extending @JamalS answer, the quoted result of Alexakis & Schlue means that any time-periodic solution must be non-asymptotically flat or non-vacuum.
For example, it is easy to construct a pp-wave solutions which would be periodic with a suitable choice of time coordinate.
The resonant properties of anti-de Sitter spacetime also provide possibility to construct time-periodic non-black hole solutions, which are asymptotically AdS. Though no explicit metrics for such spacetimes are known presently, a paper 


*

*Horowitz, G. T., & Santos, J. E. (2014). Geons and the instability of Anti-de Sitter spacetime, arXiv:1408.5906.


provides us with a numerical construction of such solution, a geon, with a helical symmetry.
