Not the US.
Dick Cheney collaborated with US major oil companies in a plot to at least take over operating the oil production in Iraq, OPEC’s second largest producer and exporter, if not get to own the oil itself outright (which has not happened as oil in the ground was and remains owned by the Iraqi government, which is the way it is in pretty much all OPEC members). Of all people, Juan Cole and many other progressives agreed that the war was all about controlling Iraq’s oil. So the US overthrew Saddam Hussein, but then what followed was civil war and discombobulation, and oil production was seriously disrupted for a long time, with those US oil companies not getting any business for a long time.
Of course, Donald Trump has repeatedly argued that the worst thing about the Iraq war was that “we did not get thee oil.” Of course, the only reason he has left any troops in northeastern Syria is that somwbody told him there is oil there and he should leave troops there to keep terrorists from getting at the oil. So there we have US troops occupying some of these wells, although there is basically no way they will ever be operated by US companies, much less owned by them. Trump is deluded if he thinks “we have got” that oil.
So what about now? According to the Iraqi Oil Ministry, there are 23 foreign corporations operating in the Iraqi oil sector. Four of these are Chinese, three are Russian, and one is American: Exxon Mobil. Last year Exxon Mobil reduced its workforce in Iraq due to security issues.