As much as I love RPGs, no, it's not necessary or even beneficial for every game. For example, there are countless action and arcade games that have no permanent character progression (though many have temporary powerups). Consider classic games like Ninja Gaiden, Contra, Castlevania, Streets of Rage, or Super Ghouls n' Ghosts. Gameplay satisfaction in games like these primarily comes from building mastery and overcoming increasingly difficult challenges. Beating them is memorable and feels great. Of course some games in this style do have ability / stat progression (e.g. Megaman and especially the X series), but it isn't required.