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The same tests any other applicants must take. Home schooled children can be at a disadvantage with large universities, however, simply because greaduating at the top of your class doesn't mean much when you were taught at home.
But in the end, how well you do on an entrance exam, and for many of the better universities, SATs and ACTs are a bare start, determines a LOT. Try taking the Harvard entrance exam, and you'll realize instantly there are no dumb Harvard students. The same is true fore most universities of this stature.
But for ordinary colleges and universities, an SAT or ACT is what colleges base everything on, home schooled or not. Score too low on an SAT or ACT and you may still get in, but on a trial basis.
But, seriously, anyone who can't score fairly high on either exam has no business in college, and probably won't do well once there.