in order:
1. Wii
Super Mario Galaxy
It's hard for me to not list Wii as first. It has VC, tons of great retail software, plenty of great DLC, and when I think about the system that I've had the most flat out fun with, Wii takes it hands down. To top it off it has IMHO the best entry in one of gaming's biggest, best and most historic franchises.
2. SNES
Chrono Trigger
SNES if you take out VC, Wiiware, and the fact that today's consoles are just superior to the ones of old, SNES wins hands down. Best library, still many unparalleled worlds, and playing Chrono Trigger for the first time to me was when I saw exactly how cinematic video games could be, a full 2 years before FFVII hit the scene.
4. Dreamcast
PSO/Shenmue
The Dreamcast was obviously the start of the next-generation despite what Sony would have you believe. One of the greatest software launch lineups to this day, a measurable leap in graphics over the previous gen, and features and overall functionality that wasn't matched until a full generation later. Four controller ports. Free online out of the box (XBX was not online until a year later and it still wasn't free out of the box thanks to Live). Voice enabled games. DLC. To this day it still stuns me how far ahead of its time the system was. But enough about that. The two games, yes two games, I was forced to pick are legends in this hobby. The first and still possibly best online console game, and arguably the first and definitely still one of the best sandbox games. PSO and Shenmue showed the world EXACTLY what the next-gen looked like, and behold it was VERY good.
4. Genesis
Revenge of Shinobi
Genesis was definitely not a "loser" in the 16-bit era. It had the superior versions of many games, usually had a faster feel to the games, and in the age of NES and SMS it was really the first time in my video gaming history (1979) that I felt as if the arcade truly came home. I picked Revenge of Shinobi because to this day, it was really the first time every that I felt like I was playing an arcade game at home (I hated altered beast and hang-on and space harrier II were just prettier versions of what we already had).
5. PSX
Final Fantasy VII
I almost feel guilty picking this. The system never really gripped me. I never held strong nostalgia for any single game on the system, except FFVII. I would be an absolute liar to say the first time I played FFVII that I wasn't blown away. I would be a liar to say that I didn't waste the first 14 days of owning it to the tune of 75+ hours beating every last nook and cranny of the game. I teared up when Aeris died, I stood in awe of the cutscenes, I rofl'ed at Barrett's dialogue, and I was stunned at the visuals (And length) of the final fight. The PSX as a system was decent and I had my fun with it, but it sits in my top five solely because FFVII will always stand out as one of the defining moments (for better or worse) of my gaming hobby.
Runner-Up
GBC
Pokemon Red
I see other people listed handhelds but I don't really consider them consoles. Still, I would be remiss to not include the GBC, or more appropriately Pokemon Red which accounted for around 130 hours of my life back in 1998. IT was my first real venture into handheld gaming (excusing very brief stints with GBs, a Lynx, and borrowing some Game Gears). Needless to say after having my GBC and wormlight glued to my hand for the better part of a month, I never disrespected the power of handheld gaming again.
edit - oh shit.. we're talking systems, not consoles? Well fuck, in that case my DS is the greatest system of all time. Best library ever, some of the best games ever, and it possibly changed the landscape of gaming more than any other system ever save for maybe the Wii. Defining game? Nintendogs. Anyone who says otherwise is a liar. Nintendogs single-handedly turned the DS from a neat system with a bit of potential into the fastest selling video game system of all time. DS sold 2 million units in one month, and it never would have done that without Nintendogs in 2005.