I don't usually declare a 'favorite game' but the games that left the longest/strongest impressions on me were Chrono Trigger, Shadow of the Colossus, and Shin Megami Tensei: Persona 4.
-Everything about Chrono Trigger is worth praise; it has incredible art direction, balanced difficulty, deep gameplay mechanics, a story that's a bit deeper than your usual SNES title, some gameplay variety and replay-ability, and not to mention one of my favorite gaming soundtracks of all time by my favorite gaming composer.
-Shadow of the Colossus was the game that got me to finally play PS2. This is one of the incredibly rare titles that I would deem as "art." I love how the setting and atmosphere tell the story so well, not to mention how well the music successfully drives the emotions of the gamer. The controls definitely has a learning curve, but when mastered, proved to be incredibly intuitive. The colossus design is beyond excellent, from their natural animations to (for the most part,) their largeness that accentuates just how small Wander (the protagonist) is. The strategies required to defeat each colossus are unique and intuitive, especially in the fights that utilize Agro (your horse). Many elements of this game can be analyzed on a deeper level, (personally, I think certain colossi are symbolic, especially the final colossus, which could represent the last of Wander's humanity; not to mention the themes of light and darkness, with more and more dark shadows crowding Wander and white doves surrounding Mono as the game progresses), which I think shows the potential of making games with a sort of literary merit.
-Persona 4 did everything that a sequel should do; every aspect of the gameplay improved and the game felt entirely fresh and independent of its predecessor. Atlus deserves a ton of praise for how maturely the game's story was written. The characters had consistent, believable personalities and ended up being round characters, changing according to their experiences throughout the game. The excellent characterization brought a new depth to the story, exploring themes that are normally almost taboo for video game writing. This is one of the few RPGs where the player can actually 'Role-play,' splitting the game between combat and student life. Both gameplay modes tie in together surprisingly well, where the friendships developed help unlock new personae and adds new combat abilities for the major characters (ex: once you hit a certain level of friendship, a major character will take a dying hit for you if you are low enough in health.)
I've played many great games in my life, but I do not judge a game solely based off of the experience from beginning to end; I also judge a game based off of the amount of time I am still awed by a game. I have been in love with these games long after I put the controller down, which was minimum 2 or 3 years for each game I listed. I honestly cannot declare a favorite though, as my opinion would probably sway each time I take an interest in a different genre.
*I know that my post is long; I tend to found my opinions on many many factors.*
