Thanks for all answers;-)

Who does he like to play with? [best choice] Whom does he like to play with? [very formal] With whom does he like to play? [very formal] "who he likes to play with ...." as it stands cannot be a direct question because it has no auxiliary verb to form the interrogative. However, it could be used as an indirect question: "she doesn't know who(m) he likes to play with, but ...." Whom sounds a bit too formal, but not grammatically wrong. It would be WRONG in the following instance to use the auxiliary verb: "she doesn't know who(m) does he play with ...." because this is an indirect question and the auxiliary verb 'does' is not required in indirect questions.

"Who does he like to play with?" (modern English) OR "with whom does he like to play with?" (old fashioned). Does this make sense?

Hello, Ms. Garcia: Nowadays a few (very few!) people still follow the rule: WHOM does he like to play with? With WHOM does he like to play? (Who = quien) (Whom = con quien) I think that most Americans would use WHO in your question. So if you wish to follow the "old-fashioned" rule, use "whom"; if you wish to speak and write as do most Americans, use "who."

Who does he like to play with?