The only way you could do this is have groups of swiss-system with the players in each group being exactly the number of games (minus 1), and setting it to not play the same player twice.
It's hard to predict sign up numbers, and those who set the tournament up can't change a single setting once another person signs up.
Example, an Azul round-robin, but one at a time (Azul chosen as it's popular, so if you make the maximum sign ups low, then you can probably predict you'll hit maximum sign ups:
* Azul
* Group stage
* Swiss system for both stages
* 2 players per game
* Max sign-ups - 56
* 7 groups (8 per group in stage 1)
* 7 games in stage one, don't match each player twice
* 1 player per group goes through - 7 players in stage 2
* 6 games in stage one, don't match each player twice