The greatest underdogs in history IMO, is not even a country, nor did they come close to having the power of a country, but they faced off against the greatest empire of their time.
Unlike countries like Vietnam which had a good amount of population and help from a Superpower, this was just a small group, and most of the time left to fend for themsleves.
The underdogs I'm talking about is a Knightly order, The Knights of the Order of Saint John (Knights Hospitaliers).
Numbering no more than 600 at any given time, three times with the help of a only few thousand militia and soldiers to bolster them, three times they faced of against the full might of the Ottoman Empire to either victory or a stand still, against tens if not hundreds of thounsands of soldiers. Most of the time they had either been written off or abandoned by the rest of Europe.
Theirs is an epic story of how this band of brothers held fast against overwhelming odds fighting one of the great empires in history. Culminating in the siege of Malta (1565), arguably one of the most decisive seiges in history, its loss would have given the Turks control of the Mediterranian.