Which friction? If you neglect the friction of the car with the air and road, then you still have the internal friction of the engine and drivetrain itself -- the friction between the pistons and the cylinders, in the bearings, et cetera, and this, too, will increase with engine velocity, so there is a point at which all the engine power would be going to overcoming internal friction. But I suspect long before that point the engine would be damaged by the extreme heat generated by that much energy being dissipated in the bearing and cylinder surfaces through friction.
If you ignore friction within the engine, too, then there is still an upper limit in a single gear because the engine cannot run fast enough -- the gasoline vapor and air can't get into or out of the cylinder fast enough, the centrifugal force on the moving parts starts to damage them, et cetera.
But on the other hand, if you allow yourself an infinite number of gears in your transmission, as well as ignoring all friction forces, internal and external, then I see no upper limit (other than the speed of light) for the velocity to which your engine can accelerate your car. Even the smallest engine, and even the largest car.
If you have constant power P = dEk/dt, where Ek = kinetic energy, then for plain Newtonian mechanics the derivative on the RHS is m v dv/dt. Multiplying through by dt and Integrating both sides gives v^2 = 2 P t / m plus some constant, or
\[v(t) = v(0) + \sqrt{\frac{2 P}{m}} t^{1/2}\]
which can be differentiated once to give the acceleration:
\[a(t) = \sqrt{\frac{P}{2 m}} t^{-1/2}\]
It's interesting that the velocity grows more and more slowly, and the acceleration slows down with time. You're right it asymptotically approaches zero, but not before both time and velocity both reach infinity.
Of course, to do this right we have to use relativistic kinematics, and I never touch relativity before lunch.