I like to stay at home
Is it:
Me gusta quedar en casa
or
Me gusta quedarme en casa
Your second option is the right one. Quedarse is a pronominal verb because the pronoun "se" doesn't carry out any syntactic function, it always agrees with the subject, and it's necessary for all the conjugation forms of the verb. Moreover, the meaning of the verb changes when a pronoun is added:
There are other meanings (quedar bien = te leave a good impression; quedarse sin tiempo = to run out of time, etc.), but those are tangential to the question. However, one relevant meaning is "quedar" in the sense of "to meet". From WordReference:
Mi hija y sus amigas quedaron a las seis en el cine = My daughter and her friends arranged (or: planned) to meet at the cinema at 6 o'clock.
If what you intended to say is "I like to meet [my friends] at home" or "I like to hang out at home", then "Me gusta quedar en casa" is fine. If what you mean is "I like to stay at home", only "Me gusta quedarme en casa" is correct.
bañar, not to do withgustar– Jdamian Apr 04 '16 at 07:58