What is the probability that a self-avoiding random walk (SAW) on a rectangular or hexagonal lattice is able to take more than $N$ steps, i.e. able to take more than $N$ steps before trapping itself by previously visiting all nearest-neighbor vertices?

Note of clarification (in response to Vincent Beffara's answer): What I specifically mean by a self-avoiding random walk is a walk that never revisits a vertex in the lattice. My question isn't "how long will the walk remain self-avoiding" but rather, if the walk is strictly self-avoiding, how many steps will the walk take before it "boxes itself in" and is no longer able to make any moves without revisiting a vertex.