Right as the 2008 election was reaching a boiling point, the makers of Mercenaries 2 released a bizarre expansion that let you play the game as either Barack Obama or Sarah Palin. The two would engage in typical Mercenaries gameplay, driving tanks and firing big-ass weapons at bad guys. In all honesty, if Sarah Palin had been elected, it's not too much of a stretch to say that this is how she would have dealt with Venezuela.
For whatever reason, in the third (yet first, chronologically) Metal Gear Solid game, we go back and find out that Lyndon B. Johnson was behind the Snake Eater initiative. At the end of the game, LBJ appears and hands Naked Snake the title of "Big Boss." It's a bit weird for Kojima to put the president of the USA in a game that takes place in such a fictional universe, but his original plan was even more out there: having LBJ take the place of Raiden.
One of the most popular video game cameos of all time, the ending of Bad Dudes Vs. DragonNinja featured a guest appearance from President Ronald Reagan (George H. W. Bush appears in the NES version, which came out in 1989). At the end of the U.S. version of the game, "Ronnie" enjoys a burger with the bad dudes. In the Japanese version, however, he builds them a statue in their honor. I don't know if this means the developers thought that Japanese people didn't know what hamburgers were, or that U.S. gamers didn't know what a statue was.
If I were to ask someone to name a presidential cameo in a video game, 9 times out of 10 they're going to name the mother of all video game cameos. Players who enter a special code can play 2 on 2 hoops as the First Couple of the 90's. Though the 16 bit graphics of the day don't accurately represent their looks, Bill Clinton's sick 3-point shot made it to into the game intact.