In the state of WY, what are the laws surrounding a person who was invited in to spend the night and has never left. There are only 3 people on the lease. Can they legally ask for rent for this person? What do the laws say? Is this their permanent address after a certain number of days?

How long has this been going on? There is no set time that a guest becomes a resident. It depends on circumstances. So we need to know more.

What do you actually want, rent money or get this person out. Both are possible.