Can you lie on a marriage license?
According to most state and country laws, it is illegal to lie when you are filling out your marriage license application. However, the lie does not necessarily invalidate your marriage.
What happens if your marriage license is not recorded?
In most cases, you are still considered married without registering the license. For example, California law requires the officiant to return the marriage license to the county clerk or recorder within 10 days of the ceremony. If, for some reason, this does not happen, you are still married.
How many times are you legally allowed to get married?
In the U.S., you can get married to one person at a time. Now so long as each marriage is dissolved (by death or divorce), there's no limitation on the number of successive marriages anyone can have. (I even know a couple people who married each other more than once, having divorced in the interim.)