"He likes her the most."
Translation:Sie mag er am meisten.
"She likes him the most" would be "Sie mag ihn am meisten". Independently of the word order, "sie" is nominative and accusative, whereas "er" is only nominative (being "ihn" the accusative". So only "he likes her is valied", regardless if you wrote "Sie mag er am meisten" or "Er mag sie am meisten". Remember! Word order is quite free, a long as you keep the verb in second position...
it is quite confusing in this case, because "sie" doesn't change form in the accusative case (it changes to "ihr" in the dative, though).
To make it clear, "he likes her the most" can be "er mag sie am meisten" (norm. order) OR "sie mag er am meisten"(abn. order, but still correct), but "she likes HIM the most" would translate to either "er mag sie am meisten" (normal) OR "sie mag IHN am meisten" (abnormal, but correct).
Because, in the accusative case, only the masculine changes form (from "er" to "ihn" ... to even "ihm" in the dative case!)