Depends on what you like.

My top 10:

Fear
Fear 2
Quake
Doom
Unreal
Alan Wake
Far Cry
Shadow Warrior (2013)
Trine
Enclave

1. Deus Ex GOTY
2. Fallout 1
3. Jagged Alliance 2

I can really only recommend games I've finished and have engaged me from beginning to end. Given I don't play that much and am fussy about what I find worthwhile that doesn't leave much.

Risen
Saints Row The Third
Guacamelee! Super Turbo Championship Edition

Currently working on finishing:

Risen 2
Two Worlds 2
Shadowrun Returns

Planescape Torment
Baldur's Gate 2
Fallout 2
Gothic 2
Space Rangers HD
Settlers II
King of Dragon Pass
Fantasy General
Jagged Alliance 2
Deus Ex

Of course I had to leave out at least 20 or 30 games that are really just as must have as these.

ITT: Everyone gives their opinion of what constitutes their own personal top 10 GOG games.

OP, dear Shyguy -- my anti-Mario brethren, its kind of pointless to give recommendations without really knowing what your preferences are. Are you looking for a list of "must-own" retro classics, or do are you also interested in newer games? Are you looking for western RPG or JRPG? Do you like point & click adventures? Do you prefer management sims? Are you more of a sidescrolling action platformer fan? Maybe something like a best of each genre sort of list?

Depending on your answers to these questions, you will have dozens of vastly different lists and each one would be accurate since you give no criteria.

this is a hard question, 10 is not many

my 10:

The Longest Journey
Interstate '76 (even though it's not the most well-adapted game in the catalogue, it's a must-play)
Planescape: Torment
Fallout 1+2
TIE Fighter
Gabriel Knight 2
Arcanum
Gothic 1+2
Thief 1+2
System Shock 2

HAH, I put more than 10~

It's good you only ask for 10, because, you should really pace yourself with your acquisitions. Wait for sales, they're pretty frequent, though, the Star Wars, D&D, and a bunch of other great titles (you can ask around) have already been on sale in the past 6 months, so, if you're itching for any of those, it might be good to just go ahead and get them because who knows when they'll come on sale again (the D&D comes pretty frequently, SW has only been on sale once, but they also haven't been here that long, but probably won't go on sale again for a while, my guess would be holiday times.)

Easy Enough;

1. Deus Ex
2. Morrowind
3. Evil Genius
4. Hotline Miami
5. Hotline Miami 2
6. Farcry
7. Kerbal Space Program
8. FTL
9. Startopia
10. Hitman 2: Silent Assassin

Apparently they are:
1. Planescape: Torment
2. Baldur's Gate 2 Complete
3. Neverwinter Night Diamond
4. Dungeon Keeper 2
5. Baldur's Gate: The Original Saga
6. Theme Hospital
7. Heroes of Might and Magic 3: Complete
8. Neverwinter Nights 2 Complete
9. Sid Meier's Alpha Centauri: Planetary Pack
10. Icewind Dale Complete

Morrowind
Prince or Persia Sands of time
Deus Ex
System Shock 2
Giants citizen Kabuto
Master of Magic
Covert Action
Fallout (1 or 2)
AvP
Carmageddon (1 or 2)

BUT
people should get more than ten games (and should also get FTL, Kerbal, Penumbra, Gothic 2, Worms, Stalker, and, you know, some others)

Throw a dart. GOG has assembled the Video Game Canon here. Half the catalog should probably be played, based on your genre preferences. There aren't a ton of mistakes in the list of games 8+ years old. Pick and choose more carefully from the indies (but play FTL), especially the Day 1s, based on your likes and incoming reviews, but their presence here is what gave GOG the relevance to secure Disney, etc.

It's a fun question. You could have 20 or 30 people post lists that were 100% unique and they would be hard to argue against.

You can go through my list of must haves
1.Torchlight
2.Torchlight
3.Torchlight
4.Torchlight
5.Torchlight
6.Torchlight
7.Torchlight
8.Torchlight
9.Torchlight
10.yeah you guessed it right.Torchlight

or you may prefer this list:
1.D&D classics with mods.(You may try new remakes and Pillars of Eternity but only if you have few extra bucks.)
2.HoM&M 3 complete (you can't find it anywhere but here. stay away from HD remake.)
3.S.T.A.L.K.E.R series with mods.
4. Morrowind GOTY with mods
5. The Witcher series(if you visit to gog once a day you may find 1 and 2 for free as it was free on various occasions.)
6. Giant Citizen Kabuto (never got a chance to play it.)
7.Deus Ex (It's not optional. If you want to be here then you must own it.Stop laughing. I am serious.)
8.Fallout classic series (it here again! And if you like this then go grab Wasteland 2)
9. Nox™ (original has ™ in its name.)
10.Torchlight (because it's the game which brought me here.)

Hmmm. Let's see what my top ten GOG list would be as it does change here and there over time. Still tough to really narrow it down.

- Master of Orion 2
- Stronghold
- Star Wars: Galactic Battlegrounds
- Doom
- The Longest Journey
- Flatout 2
- RollerCoaster Tycoon 2
- Fallout
- Rayman Origins
- Descent

Honorable mentions/Runners-up:
King of Dragon Pass, Star Wars: Dark Forces, Star Wars: X-Wing Alliance, Crusader: No Remorse, Duke Nukem 3D, Wing Commander: Privateer, Gemini Rue, Alpha Centauri, Prince of Persia: The Sands of Time, Little Big Adventure, Dungeon Keeper... and the list goes on and on. ;)

At the moment, in no specific order:

1. Dead State Reanimated
2. Shadowgate
3. Xenonauts
4. Space Colony
5. Sid Meier's Alpha Centauri
6. Soldiers: Heroes of World War II
7. Tropico 3 Gold
8. Wizardy 8
9. Star Control
10. Subwar 2050

Really 10 is not enough, I could easily have included: Myst, Tex Murphy the Tesla Effect, The Journeyman Project, Desperados, Commandos, Nexus The Jupiter Incident, Men of War, Master of Orion, Shadowrun Returns, Star Wolves...

Freespace 2

If you had to buy a single space combat sim in your life, it would be this one. Critically acclaimed, big community, source port ton of missions, mod, campaigns and total conversions available. Can be played with mouse + keyboard. Also, it's one of the first games ever on GOG.

Comparison of Freespace 2 1999/2014:
https://www.youtube.com/watch?v=199DV_UY9Hc