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](http://www.csc.kth.se/utbildning/kth/kurser/DD2440/avalg14/TSP-JohMcg97.pdf).)

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.
Incidentally, the observed running time grows subquadratically.

(See also the follow-up MO question, 
"[Travelling Salesman Problem: Can the nearest neighbor algorithm be n times longer than the optimal solution?](http://mathoverflow.net/q/210175/6094).")