I'm trying to figure out if "other than the Republicans and Democrats" means you already know that the Republicans were a third-party at one point and were just so gosh darned successful at it that they destroyed one of the previous two-party parties (the Whigs).

Political scientists often only credit third parties with having some role in framing the conversation or setting the terms of the debate (at least in US political history). Third parties are frequently organized around single issues, or a set related issues, and once they gain enough support to push these issues into the political conversation, they can be co-opted by one of the two major parties - this can especially be seen in the mid 19th to early 20th century - there were the Free Soil, Know-Nothings, Greenbacks, Labor, Populist, Progressive, and more.

A major exception to the platform getting co-opted is really the last "successful" third party campaign in the US - Ross Perot's 1992 campaign, in which he got almost 20% of the vote (and arguably threw the race to Clinton). Both Bush and Clinton were pro-NAFTA (though the treaty was divisive in both parties), so there was space for a candidate virulently against it (though as a single-issue, it obviously wasn't enticing enough to push Perot anywhere near the edge). When Perot ran again in 96, he had organized the Reform party around himself, however, the economy had improved and NAFTA and the "great sucking sound" south of the border seemed to just be boogeymen - Perot only received about 8% of the vote (though this is more than any third party candidate since).