Why are there no asteroids or meteoroids with relativistic speeds? Cosmic rays can have energies going into the $10^{20}$ eV domain. Asteroids and meteoroids originating in the solar system are probably limited in their speed because they all started out from the same lump of matter having more or less the same speed, but what about rocks from other galaxies ? Could they reach relativistic speeds in relation to the solar system? Are there astronomical events that could be interpreted as a collision between such an asteroid and a planet or star?
Just a mole of hydrogen atoms with that kind of energy would have three times the energy of the Chicxulub impact event, so I hope there's a reason for them not existing!
 A: First, the speed of other galaxies isn't too helpful. For example, the radial velocity of the Andromeda galaxy relatively to us is 300 km/s, i.e. 0.1% of the speed of light only. Moreover, internally, everything in that galaxy moves by pretty much the same speed and is confined to the vicinity of that galaxy which makes us pretty sure that no piece will reach us before Andromeda galaxy will.
More importantly, macroscopic systems in outer space aren't moving with the same huge speeds near the speed of light as the cosmic rays essentially because of the second law of thermodynamics.
When cosmic rays are accelerated to high speeds, we may treat them statistically and the high energy of these elementary particles may be assigned a high temperature. But the physical systems with many degrees of freedom prefer to evolve to the most likely, high-entropy configurations. That's why the excess energy (e.g. in a supernova) tends to divide between the elementary particles chaotically. 
In particular, the individual particles' kinetic energy results from velocities that have a random direction. At these huge temperatures (kinetic energies per particle), the atoms are unbound (well above the ionization energy) and macroscopic matter doesn't exist. So the likelihood that a large object will move towards the Earth at a near-luminal velocity is as unlikely as the possibility that the numerous atoms in the large objects are assigned velocities with the exactly equal direction although the directions are being chosen from an isotropic, random distribution. 
After some time for "thermalization" (interactions between atoms are allowed to change the system with the conservation laws' being the only absolutely constraints), the greater object you consider, the less likely it is that all the atoms in that object will be doing the same thing. This is a form of the second law of thermodynamics.
The previous paragraph prevents the creation of "coherent macroscopic cosmic rays", macroscopic objects that would move in the same direction, from the thermal chaos of hot environments such as the supernova. But even if some astrophysical process managed to eject a chunk of matter at these high speeds, the previous paragraph will still guarantee that we won't receive it on Earth. Instead, the individual atom of that speedy object would still have some residual mutual velocities so the object would split into individual atoms and we would observe cosmic rays only once again.
I could summarize the situation in this way: to shoot a large object by a huge speed from a very distant celestial body to the Earth requires one to have the same and huge radial velocities of all atoms but almost vanishing relative transverse velocities. The likelihood of that goes to zero exponentially in any thermal environment.
If one assumes that the source of the initial speed is not thermal, then one must accept that the projectiles will derive their speed from their broader environments – speeds of astrophysical bodies that already exist – and those are simply of order 0.1 percent of the speed of light as in the Andromeda example or lower. These modest speeds boil down to the inhomogeneities during the structure formation and, ultimately, to inflation, or to the planetary speeds derivable for a star. Whenever you want to locally violate the modesty of the speeds, e.g. by a gravitational collapse, you don't "shoot" any new particles before the collision and when the collision happens, the excess energy of the collision also inevitably creates a high temperature and we're back to the previous paragraph.
So near-luminal macroscopic bodies aren't observed. I would even go further and despite my being a believer that life in the Universe is extremely rare, I would say that an object moving by a near-luminal speed would prove the existence of an advanced civilization. I should be a bit careful: the gravitational slingshot is a process that allows the speed to be enhanced even naturally. But even if the source of the speed were a gravitational slingshot and the speed would be really high, like 99.9999% of the speed of light, chances would be high that some intelligence was behind the optimization of the gravitational slingshot because it's extremely unlikely for such an outcome to occur naturally.
A: Why don't we observe any relativistic asteroids? 
The answer to this question would not be complete without mentioning the virial theorem. 
Considering our galaxy as a system of $N$ gravitating objects, according to the virial theorem, twice the average total kinetic energy of all objects, plus the average total potential energy of these objects, adds up to zero. In simple terms this means that on average all objects tend to move with speeds of about 0.7 times their escape velocity. 
As we, and any objects in our proximity, are bound to the Milky Way with an escape velocity of about 550 km/s, typical kinetic velocities have values of roughly 390 km/s. That is much smaller than the speed of light that is close to 300,000 km/s. Moreover, everything around us moves roughly in the same direction, following a circular flow pattern around the core of the Milky Way. So relative velocities between objects that encounter each other are typically smaller than the viral theorem suggests. Asteroids tend to hit earth's atmosphere with speeds of about 20 km/s. The fastest impact ever observed occurred at less than 30 km/s, or less than 0.01 % of the speed of light.
In principle you can consider the Milky Way and galaxies like Andromeda that are gravitationally bound to each other (the local group of galaxies) also as one system to which the viral theorem can be applied. This gives only slight changes in the gravitational binding, and hence slight changes in the average kinetic energy.
Beyond the local group, galaxies and all objects therein are speeding away from us due to the Hubble expansion. 
A: Relativistic rocks from other galaxies are in all probability out there.  They'll never hit us, though, because they're far away and heading in the wrong direction, just like everything else we can see that moves at those velocities.
A: A macroscopic body at over 900 km/s is a very unlikely event in the Solar system (may I skip explanations?), but an astronomic body “happy” enough to collide with a neutron star will certainly impact the surface at very high speed (values depend on mass of the star). BTW, relativistic collisions between neutron stars, or of a neutron star and a black hole, are a long-researched topic in numerical general relativity.
In spite of neutron star’s strong gravity, I do not expect so many bodies with high chances to crash into the star to make such impacts a frequent event. There should be few “relativistic asteroids” in the system of a neutron star at all. Planetary systems around neutron stars are known (see https://en.wikipedia.org/wiki/PSR_B1257%2B12 ) although rare, but even if there are many rocky bodies orbiting the star, most of them should have pericenters too far from the star, whereas a close encounter is necessary to gain a relativistic speed. I do not know typical conditions to infer how quickly can such orbit decay to a lower orbit with higher speed.
