Do asteroids have a maximum speed or do they keep accelerating

Asteroids are tumbling in space and spinning around and an interesting thing is that they're all spinning rather slowly.

That is consistent with them being loose piles of rubble. They're not solid objects.
When meteorites come to Earth, they tend to travel pretty fast, around twenty kilometres per second, although the maximum is seventy. The net result of that is that they make this wonderful fireball effect. If a big one comes, the atmosphere doesn't slow it down, and that's when trouble happens. It will come all the way down to the surface and explode. On eof the nest studied meteor craters is in Arizona and it's just over a kilometre wide. The object there was about thirty metres wide.
The estimate for the asteroid that wiped out the dinosaurs is ten kilometres.

in a strict sense, EVERTHING that orbits the sun is accelerating. towards the center of mass of the solar system, by definition. that point is always within the sun, IIRC, but not usually right at the center of the sun. it is usually shifted towards Jupiter a little.

regardless...
if the eccentricity of its solar orbit is non-zero, then the SPEED of ANY asteroid is probably highest at perihelion and lowest at aphelion; that is, the asteroid moves fastest when closest to the sun and slowest when at the far point of its orbit. AFAIK, all asteroids have a non-zero eccentricity.

All things have a maximum speed. But not all things will reach it. For example, when you drop a book from your desk top, that book has a maximum terminal velocity, but it'll hit the floor long before reaching that maximum.

Asteroids are no exception. They will have a self-limiting maximum speed. And those limits might vary from asteroid and place to place in space. For example, an asteroid finding itself in a heavy dust cloud will be limited to a lower maximum than a similar asteroid in deep space free of dust clouds and debris. And of course asteroids that enter our Earth's atmosphere will be brought to a sudden fiery halt by air resistance. So it's maximum will be that moment it enters sufficient air density to start braking.