Being a gamer for quite some time now it is quite hard to recap all those moments. I think I will just name a few here:
Pokemon Blue/Red/Green/Yellow/Gold/Silver/Ruby/Sapphire/Emerald/Fire Red/Black/White
Platform: NDS
Reason: Bought a GBA just for Pokemon Ruby. After pumping it over two hundreds hours of gameplay (yes I'm not joking here) and countless battery changing the game catridge failed to work on one afternoon for unknown reason. For all the time I was playing Pokemon series (up until Emerald) I always thought that a Pokemon at LV100 would be the best. However when I'm starting to get used to Battle Tower in Emerald, something popped in my mind thinking about the game mechanics so that I can further fortify my teams. As it turned out there are tons of elements that need to be considered when it comes to competitive battling such as breeding, Effort Values (EV), nature, hidden power. Spent months learning about them on Serebii and Smogon University. Few months later learned something about a hacking software called PokeSav which can practically modify a Pokemon attributes according to ones' heart content.
Metal Gear Solid 4: Guns of the Patriot
Platform: PS3
Reason: Remembered breaking my PS3 controller, literally speaking, by throwing it on the floor after an hour of failed attempt to go through a motorbike riding/chasing scene without dying, which is a requirement to get the Big Boss emblem. At the end I managed to get it. It was a very satisfying experience. It's a shame there is no achievement trophies in the game, else it would be a huge bragging right.
Persona 3 FES & Persona 4
Platform: PS2
Reason: Have heard of Persona series before but never bothered to check it out. Few months ago decided to give FES a try since no more PS2 games to play. It turned out to be one of the best PS2 games I had played last year and I checked in almost 70+ hours of gameplay time. I even study about tarot card thanks to this game as I thought the name used is just some random names. For Persona 4, there are some mixed feelings towards it based on the reviews I read, some praise it while others hate it. For me it doesn't matter as I am interested to see what does (or CAN) the sequel offer when the prequel was already at its best. It turned out that the game graphic for P4 had improved dramatically with a much vibrant color (yellow being the prominent ones) and a better battle system. It's a shame that there isn't any post game content for P4. Thankfully there are multiple endings for this

One thing I regret for playing P4 is that now I can't even look at P3 graphics!
Dragon Quest VIII
Platform: PS2
Reason: The first cell-shaded game I played on PS2. And it leads me starting to search and play other cell-shaded games, e.g. Borderlands (PC), Naruto Shipudden Ultimate Storm (PS3), Naruto Ultimate Heroes 3 & 5 (PS2). I easily checked in 100+ hours of playtime due to the post game content, Dragovian Trial and Monster Arena are very memorable.
Etrian Odyssey I / II and III
Platform: NDS
Reason: First introduced to this game while searching for 'best ds rpg'. I can still barely remembered one of the review I read, which praise the game for being one of the most difficult game to play on DS. At first I doubt to even pick up this game because of the screenshot, which features a simplistic battle scene without any character model or animated enemies (think Final Fantasy series). But at the end decided to give this game a try and soon starting to love it. Too bad there is no news for the fourth installment.
All Final Fantasy series
Platform: SNES, DS (FFIII and FFIV), PSP (Dissidia, Crisis Core, FFVII), PC (FFVII, FFVIII), PS2 (FFX, FFX-2, FFXII), PS3 (FFXIII)
Reason: One of the elements that I'm looking forward in each series is the battle scene, which is typically the most appealing element for any JRPG games aside from the plot. For me, even if the plot is cheesy but features a sophisticated or good battle scene engine, I would be glad to play through it. It's hard even for me to retell the story for all this series without having to play through them again, aside from some title such as FF7, 8, X, X-2, XII and XIII. I can barely remembered the storyline for SNES series anymore

Now for some break down:
FFVII: Having played this game twice due to the attachment that I felt in this game. I believe this I am not the only one here as many, even pro gamers had replayed this game countless of times. And mine is only two.
FFVIII: The most significance elements I remembered is the summoning sequence, which is at its best at that time of year. The Eden summoning sequence is very very long indeed, but awesome.
FFX: I played this game about one year after finishing FFX-2 just out of curiousity. Playing this game make me recap the moments that followed in FFX-2. It's good to see Yuna being able to summon!
FFX-2 is the first game I insert in my PS2 when I hook it to my TV for the first time. Heck that is the reason why I purchase a PS2 at the first place!
As for FF13, many critics had said that the storyline is pretty linear, which I highly doubt it as it's by Square Enix, just look at their previous titles! However upon playing the game I felt the reputation that I had in mind had been screwed over the edge. If not because of the game graphic I would instantly sell of the game after the first hour. Serious. I keep thinking when will this game ends when playing it. Well at least that is what I thought until I reached the mid game. The post game is pretty fun too. Gil hunting on giant turtles is truly priceless moments. Keep crossing fingers to earn those Platinum thingy (forgot the name but still remembered that it could sell for 150k, or did I get this wrong as well XD). I don't think I will play FFXIII-2 any time in the future.
SE, I salute you.

Mass Effect 2
Platform: PC
Reason: The best game by Microsoft that I played on a PC. I like the intergalactic species, planets exploration, and the mixing of RPG, adventure and shooting genre. The way that this game offers multiple dialogue options (humor included) and consequences that follows are pretty interesting. I dislike the options offered by Dragon Age as most of the time it is hard to choose what is felt right or wrong. I'm not kidding as the selection sentence is pretty long in DA.
Disgaea 2
Platform: PS2
Reason: The first Atlus game I played on PS2 that offers deep tactical thinking and is most difficult turn based games I had ever played (FF Tactics feels like child's play when compared to this game). Treasure map and pirate hunting are the things I can think of. I even spent weeks of countless late nights just to search for those treasure maps.
There's just much more to list down but those listed are just the good ones, or what I could think of at the moment of writing