My list would be very short, and somewhat archaic, if I only included the games that I've played the actual board game. Of the modern "European style games" I own very few other than Carcassonne. I have however, like Hounk, been playing a lot of different games on Yucata.de, some of which I would like to purchase the 'cardboard' versions. Also, I own almost all of the Carc spinoffs, so they should probably be on a separate list.
So, for games that I actually own (considering all the versions of Carc to be just one game) my list would be as follows:
1. Carcassonne
2. Catan
3. Spades (card game)
4. Briscas (a Puerto Rican card game that I believe originated in Italy... known as Briscola in Italy)
5. Cribbage
6. Struggle for Catan
7. Poker (most varieties)
8. Phase 10
9. Cards Against Humanity
10. 10,000 (Dice game)
There are probably others that I have that I haven't played in a while or even forgot about.
Of the Carcassonne games, this is my top 10:
1. Carcassonne base game + expansions
2. Carcassonne The City
Carcassonne Gold Rush
Carcassonne New World
Carcassonne Discovery
6. Carcassonne The Castle
Carcassonne South Seas
Carcassonne Hunters and Gatherers
Carcassonne Star Wars
Carcassonne Ark of Covenant
The only Carc game so far that I haven't liked is Hill and Dale. I'm looking forward to Amazonas.
Of games I currently play on Yucata.de, I like the following:
Carcassonne Hunters and Gatherers
Carcassonne South Seas
Castles of Burgundy (just learned this a few weeks ago and it has become one of my favorites. I am looking to buy this one.)
Nations the Dice Game (would like to own but it seems rare and very expensive)
Roll Through the Ages
To Court the King
Jaipur
Call to Glory
Cacao
Automobiles