Here is the full list of all 13 Mahjong Royal Towers trophies - there are 9 bronze, 3 silver and 1 gold.
Finish the first level with three stars
Achieve the x5 multiplier
Score 100000 points
Collect 30 crowns
Finish the Tournament Field
Finish the Castle Courtyard
Finish the Royal Garden
Finish the Refectory
Finish the Armory
Use bonus 50 times
Remove 1000 pairs of tiles
Finish the Throne Hall
Finish 30 levels with the best result