A careful experimental analysis of the nearest-neighbor (NN) heuristic (among other heuristics) is described in this paper:
Johnson, David S., and Lyle A. McGeoch. "The traveling salesman problem: A case study in local optimization." Local search in combinatorial optimization 1 (1997): 215-310. (PDF download link.)
They find that the growth rate, in comparison to the Karp-Held lower bound, "appears to be proporotional to $\log n$." This is the theoretical growth rate w.r.t. the optimum. In particular, for random Euclidean instances up to $n=10^6$, they find (Table 1) that the NN heuristic leads to paths about 25% longer than the Karp-Held lower bound
(See also the follow-up MO question, "Travelling Salesman Problem: Can the nearest neighbor algorithm be n times longer than the optimal solution?.")