Is there anything in the universe that is completely static?

The problem is that when you talk about "static" you're implying that there is some universal co-ordinate system to judge these things by. Relativity asserts that no such reference frame exists and therefore the only measures we have for motion etc are dependent on the reference frame that is watching it. Between two comoving objects they will see each other be static, but a third frame moving some other way won't see it that way and the two comoving frames will see the third move.

[by comoving I just mean two objects moving together, not anything GR related]

For a massive particle you can always find a frame where it is at rest - its proper frame, and you can always find a frame that is moving relative to that particle (so according to that frame the particle is in motion). A massless particle is always moving and has always the speed of light relative to every inertial frame. In order for the state of motion to be defined, one has to specify the frame relative to which the state of motion is evaluated.

For us to talk about "static," we have to be able to talk about a reference frame in which we are measuring something. As several people have mentioned, it is always possible to construct a reference frame with respect to a particle, and that particle will be static in that frame.

However, if I venture a guess and say that you are talking about a system which is unchanging the current prevailing opinion is that everything changes. In particular, the first law of thermodynamics says that entropy is always increasing, so there is always energy being converted into heat.

Could there be a system that is truly static? We actually can't disprove it. But from what we have seen, all macroscopic systems exhibit thermodynamic behaviors -- they all increase in entropy.

We can theorize about the existence of static systems, especially on the small scale, but to the best of our current understanding, no system actually earns the title of "static."