Some of the fastest stars in our galaxy travel at up to 700 km / sec. These are travelling so fast they will leave our galaxy. There are many more traveling at 300 km / sec. They are all moving outwards from the galactic center which is seen as evidence that they were accelerated to those speeds by encounters with the Milky Way’s central supermassive black hole. They are in bound orbits but are short lived stars which will have reached the end of their lifetime by the time they start to come in again - assumed to be 3 times the mass of the sun or more, and lifetime no more than a billion years. Annual Review of Astronomy and Astrophysics

The fastest known hypervelocity star though is US 708 at 1,200 km/s. It’s hard to explain but one theory is that it was a close companion with a white dwarf, dumped matter on the white dwarf as they spiraled together leading to a supernova explosion that blew the white dwarf apart completely leaving its companion to slingshot off at this very high speed. Hypervelocity stars wander cosmos

The very fastest stars though according to models could come from interactions of two supermassive black holes spiraling around each other before they merge. You might wonder how that is possible, but it could happen as a result of galactic collisions leading to the super massive black holes at the center of each galaxy merging. For instance in our future the Andromeda galaxy will eventually collide with the Milky Way. This would eventually lead to the two super massive black holes merging, a long way into the future.

There’s a super massive black hole in the center of each galaxy. Eventually a long way into the future they will spiral together and merge

Andromeda–Milky Way collision This will happen about four billion years from now. Stars are so far apart that probably no stars will collide, but eventually the two galaxies will merge to make an elliptical galaxy.

The supermassive black holes at the centers of the galaxies eventually merge in the process. As the two black holes spiral together before they merge (not shown in this visualization), that would lead to some stars being accelerated to vast speeds through interactions, at speeds of a thirtieth up to a third of the speed of light, which could then travel vast distances. Such stars could have traveled for billions of years across the universe already.

This could lead to stars that whip around them and get ejected traveling at speeds from a thirtieth up to a third of the speed of light. Such stars could have traveled billions of light years across the universe already. There may be as many as a thousand of these relativistic speed stars in every cubic megaparsec - but that’s a huge region, a cube over three million light years to a side.

There is probably at least one star traveling at a third of the speed of light somewhere between us and the Virgo cluster 53 million light years away, but it would be hard to detect, however we may find stars with velocities of several thousand kilometers per second in future surveys with the James Web Space Telescope etc. See The Fastest Unbound Stars in the Universe