Haven't seen this one suggested yet: Quite literally any Shin Megami Tensei game.
Super Nintendo: Shin Megami Tensei, II, and if... (SMT and its sequel have English patched versions, and "if...", a game released after II but set between the first and second games is only in Japanese)
PS1/X: Persona and Persona 2 (Persona 2 comes in different parts, Eternal Punishment and Innocent Sin; Innocent Sin is the first Persona game released internationally)
On PS2: Nocturne (Europe version is Lucifer's Call; set in post-apocalyptic Tokyo, you're forced to turn into a half-demon to survive and decide the planet's fate through many choices; not recommended for the light-hearted or narrow-minded), Devil Summoner and Devil Summoner 2 (both are hack-and-slash with a large strategy element to it), and Persona 3 (and its added-on release FES, with an after-story of the original game) and 4 (simple battle system but it works very well, both being combinations of high school sims and organized strategy fighting).
PSP: Persona Portable (remake of the original Persona) and Persona 3 Portable (added-on version of Persona 3, with a new main character to choose from).
On the NDS: Devil Survivor (very well made, with many endings depending on your choices throughout the game, and the battle system plays similarly to turn-based strategy games like Fire Emblem), and Strange Journey (a more futuristic game and the most recent of the series, marked as the fourth game of the main series).
I don't recommend ANY of these games for religious persons, as they may find the continuous use of demons (in the Persona series, they're referred to as Personae instead) offensive or otherwise blasphemous. Persona 3 I don't recommend for anyone who doesn't like guns, suicide (who doesn't?), or any form of self-harm, because you need to shoot yourself to summon your Personae in this game (it's still very fun).