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An easy way to get infinite approval is to go to any country that has US troops and is a staunch ally (britain, germany, italy, japan, etc). Enter the troop menu and repeatedly click the button to change the troop levels without altering the number of troops first; each time you do this, if the country accepts, it will raise your popularity a little (1% to 1/3% per click, less the higher your popularity is already). It even works if the troop level is 0, as long as the country accepts. I used this trick to conquer the entire world except a couple of countries I couldn't select, while maintaining 90%+ approval and altering the budget to get vast economic growth. The only feature I found that interferes with the exploit is if you mess with countries' foreign aid packages, then the exploit may stop working. Also beware of lowering social spending too much - it seems like the popularity level may get stuck at 0 if it goes too far below 0 from a single budget change. Dropping to 5% each year with successive cuts and then exploiting back up probably works, though I just stayed in a bunker whenever I unpaused the game to avoid being assassinated.