Nah, it's fine. Plenty of functions have vertical asymptotes. See for example the trig functions of tangent, cotangent, secant, and cosecant, each of which in fact has infinitely many vertical asymptotes.
What's proposed with warp 13 is that the part of the speed scale on the other side of the vertical asymptote at warp ten behaves differently from the part before. According to the TNG tech manual, warp factors occur where power consumption is locally efficient. The idea is that if the abscissa (warp factor) parameterizing warp field geometry is increased beyond ten, new optimal nodes appear, one of which is indexed by the abscissa value of 13.
I am sorry but that doesn't work. A vertical asymptote means that before you get to warp 11 for example, there is an infinity to go through first. I don't care what you do when you're at wrap eleven if to get there you have to go through an impossibility first.
Same thing about real physics. You can't get to the speed of light without an infinite amount of energy. You could argue that once you're at say twice the speed of light things get normal again but to get there you must dispose of an infinite amount of energy and that's impossible.