Well there have been several ‘invasions’ that led to a general or at least partial conquering.
Obviously with the first humans came a swarm like effect that spread over both american continents around 30,000 BC.
since then the Spanish, British and French were the next ones to largly dominate either continent.(the Vikings “discovered” the americas before them but failed to establish themselves) After the french and Indian war Britain and Spain became the only serious players in the conquering catagory. And obviously since the Revolutionary war and the Spanish Revolutions in south america, the USA has become the only piece capable of the ill-fated check mate of this continent.
But in terms of Conquering the Continent in General, it has to be the US. Every nation attempting the same as us was incapable of putting down the most difficult terrains and the native people routinely harassed them enough to be left alone. America had the advantage of time, however. Staying in on the continent for over 250 years, only Britain surpasses that in their dominance of Canada from the early parts of the 18th century to after WWII.