The closest thing to "just cross country" is probably foxhunting. No, you can't go and just do XC.
BN--WTC dressage with 20m circles and a free walk. XC--up to 2'7" high, 4' wide (at the base--top max width is 2'11), brush up to 3', up to 350 mpm. SJ--up to 2'7 high, 3'3 wide, triple bars up to 3'11" wide, 300 mpm.
That's the lowest level for recognized eventing. As verona pointed out, there are unrecognized shows (on rare occasions held in conjunction with the recognized events but more commonly held separately).
Recognized levels are Beginner Novice, Novice, Training, Preliminary, Intermediate, Advanced (in the US--it's different in other countries). You have to qualify for Prelim and above. The USEA gives very precise requirements for each level in the USEA rulebook, which can be found online. In fact, all kinds of information exists on the USEA website United States Eventing Association, Inc. - US National Combined Training, Horse Trials: Dressage, Cross Country, Show Jumping
There are sometimes derbies and combined tests. Derbies are unrecognized and typically are a dressage test followed by a course that is a combination of SJ and XC. CTs can be recognized and officially are two of the three phases, though in the common practice is for one of those phases to be the dressage phase and most commonly the second phase is SJ.