Which star / galaxy is moving away from us the fastest? I know that we have measured the rate a lot of stars and galaxies move away from us using the doppler shift, and I know that the further a star / galaxy is the faster they accelerate away from us due to space inflation. I'm wondering, did anyone categorise which star / galaxy is moving towards us / away from us the fastest? How fast are they going in relation to our solar system?
When a galaxy recedes from us, the light we see from it is redshifted. For galaxies at cosmological distances, this redshift is fundamentally different from a Doppler shift; whereas the latter is due to a velocity difference between the emitter and the receiver, a cosmological redshift is due to photons traveling through an expanding space$^\dagger$.
Hence, as @uhoh comments, your question is equivalent to asking &quot;Which galaxy has the highest measured redshift?&quot;. Redshift is arguably the single most important concept in astronomy, and indeed we do catalogue the redshift of all galaxies, if possible. For our adopted cosmological model, the cosmological redshift can be translated to both a recession velocity, a distance, and an age of the Universe when the light was emitted.
The answer is GN-z11 (Oesch et al. 2016) which has a redshift of $z=11.09$. This corresponds to a distance of $d = 32.2\,\mathrm{Glyr}$ (i.e. billion light-years), and hence, by Hubble's law, to a recession velocity of
$$
v = H_0\,d = 670\,000\,\mathrm{km}\,\mathrm{s}^{-1},
$$
or more than twice the speed of light. Furthermore, the light we see today was emitted when the Universe was only 410 Myr (i.e. million years), or 3% of its current age.
You may think that &quot;twice the speed of light&quot; violates the theory of relativity, but this velocity is not a velocity through space. Both our galaxy (the Milky Way) and GN-z11 move through space at modest velocities of a few 100 km/s. The recession is merely due to space expanding, and space is allowed to expand at whatever rate is wishes.

$^\dagger$&quot;Fundamentally different&quot; might be a too strong statement, since the cosmological redshift can be interpreted as infinitely many infinitesimally small Doppler shifts. However, a hypothetical scenario that emphasizes the difference between the two types of redshifts is the following:
If an emitter and an observer are stationary wrt. each other when the emitter emits a photon, then start moving away from each other while the photon is traveling, then stop again before the observer receives the photon, the observer would measure zero redshift.
On the other hand, if space is static when the emitter emits the photon, then while the photon is traveling suddenly expands by a factor of, say, four, and then is static again when the observer receives the photon, the observer would measure a redshift of $z+1 = 4$.