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