In General Relativity (which is the theory we have to use if we're talking about black holes), gravity is not a force, it's spacetime curvature. The spacetime curvature at any point in spacetime depends only on the matter in the past light cone of that point. Heuristically, "gravity" travels at a finite speed in GR. But there is matter and energy everywhere in the universe, so there is spacetime curvature everywhere; there is no place you can go to "escape" from its effects.

So thinking of gravity as a Newtonian "attraction over infinite distance" doesn't really work in GR, but neither does thinking of it as a "force" with a finite range. It works differently from either of those.

Thinking in continuous terms, it would be infinite and inversely proportional to distance squared. But thinking in terms of a distortion in space-time and also considering that it may be quantized, perhaps there is some limit. Also for points that are separating faster than the speed of light due to inflation gravity couldn't alter that space.

Of course, nothing can have infinite range in an infinite Universe, because the infinite sum of gravitation fields of an infinite number of objects can create infinite force at any point of the Universe.

At some range, any field will decrease to below thermal noise of medium.