I got the longest path down to
and I don't think it can be improved upon.
1 2 0 3
4 0 5 0
0 6 0 7
8 0 9 A
(The A can be any number, including a previously unused one.)
If it were only the numbers 1-9, then choosing one of the digits as "padding" and making a checkerboard with it, we could sprinkle the other digits in between, for a result of three.
However, we have an extra digit, so we must have at least one group of 2 adjacent non-padding digits. Sadly, there doesn't seem to be any configuration for fitting 9 squares in non-adjacent groups of maximum size 2 onto a 4x4 grid.
Therefore, we must have at least 1 "island" of three non-padding squares, and since we cannot isolate it completely (would need a double layer of padding), the above solution seems optimal.
(Having repeats of some non-padding digits isn't going to help, since turning one of those duplicates into the padding digit cannot make the longest path longer than it was before.)