From the virial theorem, peculiar velocities resulting from gravitational processes generally have magnitudes of the order of
$$
V \sim \left(\frac{GM}{R}\right)^{1/2},
$$
where $G$ is the gravitational constant, and $M$ and $R$ are the mass and radius of the system.
Galaxies
Galaxies have peculiar velocities of the order of several 100 km per second; small velocities for small groups (~100 km/s; e.g Carlberg et al. 2000) and large velocities for rich clusters (~1000 km/s; e.g Girardi et al. 1993).
The maximum velocities are reached when a galaxy plunges through the center of the cluster, and many clusters thus have a few galaxies surpassing 2000 km/s (e.g. Einasto et al. 2010).
Outflows
Many galaxies eject particles through galactic winds, powered by active galactic nuclei (AGN) and/or stellar feedback (radiation pressure and cosmic rays from massive stars, as well as kinematic and thermal feedback from exploding supernovae). Although these processes are not gravitational per se, the outflow velocities typically do scale with the size of the galaxy due to various scaling relations (more massive galaxies host more massive central, supermassive black hole, and have more vigorous star formation).
The particles don't always escape; typically the outflow velocities $v_\mathrm{out}$ are a bit smaller than the galaxies' circular velocities $v_\mathrm{circ}$ which, in turn, are a factor $\sqrt{2}$ smaller than the escape velocity $v_\mathrm{esc}$. For instance, Rupke et al. (2005) find $v_\mathrm{out} \simeq v_\mathrm{circ}^{0.85}$.
Hence, typical outflow velocities of gaseous material are several 100 km/s (e.g. Sugahara et al. 2017), but sometimes surpass 1000 km/s and escape (which is the reason that the intergalactic medium has been enriched with metals).
However, the most highly accelerated parts of the jets resulting from AGN often reach relativistic velocities, i.e. velocities approaching the speed of light $c$. Similarly, cosmic rays and neutrinos have velocities which are very close to $c$. These are, of course, the maximum peculiar velocities you'll find, but I suppose you're thinking of macroscopic bodies, not particles.
Stars
Stellar velocities — that is, the velocities with which stars move around inside galaxies — also increase with the mass of the galaxy. In a galaxy like the Milky Way, a fairly typical galaxy of $M\simeq10^{12}\,M_\odot$, stars revolve with velocities of 200–250 km/s, with a dispersion of a few tens of km/s (that is, in the reference frame of a given star, nearby stars typically have velocities of order ~10 km/s.
However, sometimes stars in dense environments (globular clusters) come close to each other and acquire larger velocities. And if binary stars come close the the galaxy's central black hole, it can reach very high velocities, more than 1000 km/s.
The record for such hypervelocity stars
is 1755 km/s (Koposov et al. 2019).
Asymmetric supernova explosion in stellar binaries may also give rise to runaway stars with velocities of several 100 km/s (Tauris & Takens 1998).
Solar system bodies
For the sake of completeness, I'll end by discussing planets, though they of course have smaller peculiar velocities (in the reference frame of their star). You can use the same formula as given in the top, and since most of the mass is concentrated in the center (the star), $M$ and $R$ are just the mass of the star, and the distance between the star and the planets, respectively. For instance, Mercury with a distance of 0.4 AU from the Sun has a velocity of 47 km/s.
Sometimes planets are ejected and become rogue planets, which may reach velocities of more than 100 km/s (Bardalez Gagliuffi et al. 2020).
