Ben and Mabel (Mable) cardgame? What's the "real" name?

Question

My friends play a game they call "Ben and Mabel". I believe they learned it from their Dutch brother-in-law.

This is my memory of playing it.

You play with multiple decks mixed together. Maybe two or three. Jokers included (they're wild).

You start with three cards. You play 13 rounds; in each subsequent round, your hand grows by one card. Each turn you draw a card and discard a card. You can draw from the face-up discard pile or the face-down pile. You play melds of runs in the same suit or groups of three of a kind or more. If you are initially dealt three cards, threes and are wild. If you are dealt four cards, fours and jokers are wild. Eight cards, eights and jokers are wild.

Round one is easy because there's only three cards. Round five is the hardest because you have to get five of a kind (wilds OK) or a run of five in the same suit. Once the rounds get big it gets easier too.

There is some scoring involved at the end once someone goes out.

Sound familiar?

What's the "real" name?

Answer 1

This sounds like a form of Contract Rummy.

There are dozens of variations of Contract Rummy which have different rules of how many and what size of sets and sequences are required in each successive round. In my brief overview of different variations on pagat.com, I didn't find an exact match to the clues you've given, I suggest you have a look there and try different contracts!

Answer 2

I have a commercial version of the game called "Five Crowns". It is played with a custom deck that has 5 suits, each of which contain 2 of each card from 3-K, plus some jokers. Otherwise the rules sound identical.