Candidates may be found among those countries which have retained substantially the same size and colour for a given denomination over a long period, even if at times they have fallen out of use through inflation, regime change etc. I'm thinking most of pre-euro France and Germany here, but it might also apply to various South American states. (Someone upthread mentioned Switzerland as having the oldest designs still in use; I think countries like Venezuela and Bolivia give it a run for its money.)
The French 50c-1Fr-2Fr of 1896-1918 are more or less identical to the same denominations issued from 1960 to 2001. The latter are in Ni, the former Ag835, and the former are thinner (most noticeable with the 50c), but they are similar enough I'd have thought to have passed in circulation before the euro.
The same situation arises in Germany. The 1-10pf coins of the Weimar Republic and the first few years of the Third Reich (before they went zinc) are similar at a glance to those of the Bundesrepublik. The silver 50pf, 1 and 2 Mk coins of the Kaiserreich are likewise similar enough to have passed for the same denomination before the euro. I received a 1935 5 RM coin in change in 1989/90.
Of course, this is all academic because these countries now use the euro.
Does French Polynesia still use the francs it issued in the 1950s -- those big aluminium jobs? If so, they're the same size as the French 50c-5Fr coins issued by the Vichy government and then into the 1950s. So you could perhaps find an aluminium 1Fr coin from 1941 (I think that's the first of the Al series, and it would be Vichy, not independent) in circulation.
I would agree that the winner by a country mile is the US. It has preserved the situation that used to obtain in the UK before 1971, when you could routinely find 100-year-old pennies and halfpennies in circulation. I imagine the same was true up to 1920 of post-1816 silver (which remained legal tender until much later -- 1990 for shillings -- but would have been unlikely to be used in practice after the reduction in silver content).