Is there a limit to the number of times you are able to multiclass, for example could you get 1 level in 4 different classes?
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Yes. Page 208 of PHB notes:
This is true for all classes except bard who have an explicit provision to take multiple class-specific feats:
While the windrise ports background did allow multiple multiclass feats, the recent errata eliminated that option.
To address your specific confusion, there is no such thing as taking "levels" in other classes, only feats and power-swap feats. In essence, you are always your primary class (though with a multiclass feat you can take prestige paths and epic destinies and class-specific feats for your secondary class, as the feat indicates that you count as a member of that class for game purposes.)