Too many to remember right now but some really good ones:

1. Gravity Rush (Discovery of Gravitation) - https://www.youtube.com/watch?v=XRf5XKeo-DQ

2. Transistor (Old Friends) - https://www.youtube.com/watch?v=ql8K3KJyWgY

3. Child of Light (Magna's Heart) - https://www.youtube.com/watch?v=Wew8kC4tvNQ

4. Kingdom Hearts (Hikari) - https://www.youtube.com/watch?v=AVSGMYAsCxE&list=PL2FFE386842EA290F&index=2

5. Kingdom Hearts (Dearly Beloved) - https://www.youtube.com/watch?v=ExsP_nCOiJY&index=1&list=PL2FFE386842EA290F

I have plenty, but I'm not going to show you that today. Instead, I'll bless you all with some amazing fan remakes of game music. (Mostly Pokemon) Prepare yourselves.

Super Mario World

https://www.youtube.com/watch?v=T5KBI_9bm-I&spfreload=10

-

Super Metroid

Kraid/Phantoon/Crocomire

https://www.youtube.com/watch?v=SoIcUYoo3gs&spfreload=10

Mini Boss

https://www.youtube.com/watch?v=I7v3oPyGMh8&spfreload=10

Wrecked Ship - Super Metroid

https://www.youtube.com/watch?v=oBZQYLyyxb8&spfreload=10

Brinstar

https://www.youtube.com/watch?v=BS5J9taYBBQ&spfreload=10

Brinstar (Red Soil)

https://www.youtube.com/watch?v=phDSZ-uy0d8

-

Pokemon

Gary (Blue) FUCKING OAK:

https://www.youtube.com/watch?v=L5ceozY60pQ&spfreload=10

GSC Main Theme:

https://www.youtube.com/watch?v=385Tinq70R8&spfreload=10

Johto Trainer Battle Metal:

https://www.youtube.com/watch?v=rODPdg5W35M&spfreload=10

Viridian City Smooth Jazz:

https://www.youtube.com/watch?v=SrgaUY4EDLc&spfreload=10

Team Plasma Battle Metal:

https://www.youtube.com/watch?v=gZvtawZo-a4&spfreload=10

Kanto Gym Battle (GSC):

https://www.youtube.com/watch?v=qcoGi29Bldg&spfreload=10

Sinnih Route 201:

https://www.youtube.com/watch?v=C5fnYFK-KDo&spfreload=10

Cynthia Battle Theme:

https://www.youtube.com/watch?v=l7Xr_cQXgj4&spfreload=10

Johto Trainer Batturu:

https://www.youtube.com/watch?v=OImT5TgdOBI&spfreload=10

Violet City:

https://www.youtube.com/watch?v=gUvMzxh9quI&spfreload=10

Team Rocket Encounter (GSC):

https://www.youtube.com/watch?v=ArdbL-J3Axg&spfreload=10

Route 27:

https://www.youtube.com/watch?v=QE3poMPtU_8&spfreload=10

Zone of the Enders The Second Runner, Track: Beyond the Bounds: https://www.youtube.com/watch?v=Niruwh4UwS8

Champion Lance/Trainer Red Theme - Pokemon https://www.youtube.com/watch?v=iDFEUfS4r_A
Solaris Phase 2 - Sonic The Hedgehog (2006) https://www.youtube.com/watch?v=JG4yunUYG-Y
Dr. Wily Stage - Mega Man 2 https://www.youtube.com/watch?v=N4VtxWRfPOk
You Are My Hope - Bravely Default https://www.youtube.com/watch?v=lIVBqSIVcSQ
Scorch n' Torch - DKC: TF https://www.youtube.com/watch?v=JFP4wn0dejc&index=28
Halo 2 Theme https://www.youtube.com/watch?v=tx4VFMmvOvo

My list is under the restriction of one song per game, otherwise, every song in my list would be from Xenoblade.

1. Xenoblade Chronicles: You Will Know Our Names - https://www.youtube.com/watch?v=V4vCEHpv6Hg

2. Pokemon FireRed/LeafGreen: Vs. Champion - https://www.youtube.com/watch?v=2T0pG8PKhBU

3. Super Smash Bros. Melee: Fire Emblem (Together We Ride/Main Theme) - https://www.youtube.com/watch?v=HjRmdwGILfc

4. Sonic Unleashed: Rooftop Run Day - https://www.youtube.com/watch?v=gtSmFc-hHrw

5. Kid Icarus Uprising: Boss Battle 1 - https://www.youtube.com/watch?v=VwPLsIP1AmI

There are so many.. These are some of the most recent (and a few off the top of my head) in random order:

Gran Turismo 5 - Daiki Kasho (Soul On Display)
The Last of Us - Gustavo Santalolalla (Main Theme)
The Unfinished Swan - Joel Corelitz (Main Theme)
Assassin's Creed - Massive Attack (Teardrop)
Unreal Tournament 3 - Rom Di Prisco (Menu Theme)
Super Mario 64 - Koji Kondo (Inside the Castle Walls)
Gran Turismo - Feeder, Ash, Masahiro Andoh (Sweet 16, Lose Control, Aston Martin Dealer)
LittleBigPlanet - Café Tacvba (Volver a comenzar)
Pokémon Soul Silver - Go Ichinose (Azalea Town, Route 29, Johto Gym Leader Battle)
Grand Theft Auto 2 - EZ Rollers (Short Change)
Grand Theft Auto 5 - Tangerine Dream (Loading Screen Theme)
Crash Bandicoot 2 - Josh Mancell (Sewer or Later)
Spyro the Dragon - Stewart Copeland (Opening Theme)
Need for Speed Undercover - Pendulum (9000 Miles)
FIFA 14 - Oliver (Mechanical)

And a little bonus (not from a game but I've loved it since I was a kid :p):

Pokémon: The First Movie - Vitamin C (Vacation)

Portal - Still Alive
www.youtube.com/watch?v=Y6ljFaKRTrI
Super Mario Galaxy - Gusty Garden Galaxy
www.youtube.com/watch?v=bcZhJDUFb58
Super Smash Bros. 4 3DS/Wii U - Battlefield
www.youtube.com/watch?v=PZvO3h7Wqb0
Uncharted - Nate's Theme
www.youtube.com/watch?v=kgcq_ZtqbO0
Halo Theme
www.youtube.com/watch?v=Z5yVOFokLVY

What about these

https://www.youtube.com/watch?v=SOSQeO97xoQ

https://www.youtube.com/watch?v=IUrqNEJeF4w

The songs in FF XIII

Yeah I've listened to those 2 quite a lot as well.. Soul on display is just sick imo. I'm probably being a little nostalgic too hearing it for the first time at E3 2010 while watching that beautiful GT5 trailer.