Nancy thinks that she is a better chess player than

I

Me

Myself

Nancy thinks that she is a better chess player than me (coloquial)

Nancy thinks that she is a better chess player than I am (classical)

Nancy thinks that she is better chess player than me.

Myself is doing something alone. I made the doll by myself.

I is used by an action you made or discribing yourself. I am tall. I worked alone.

Me is used to mention yourself. Natalie was playing with me.