There is now a leaderboard that will show your score.
What is the score?
The score counts how many unique items you have. It does not count things like tickets or gold. Only items that appear in the in-game inventory are counted.
Example:
2 x Stones
3 x Sticks
1 x Clock
---> Score: 3
For now it is only visible in the steam community tab. It will later be added in-game.