My sister up-cycled a Jenga game for Monkey's Christmas present this last year. It has turned out to be a really fun toy and has been used in many ways.
Our goal was to spend less than a dollar on each person's gift, I believe she already had the game and bucket.
I would keep an eye out for used games at garage sales and thrift stores, since the box is going anyways it won't matter if the cardboard is beat up.
I'm thinking she used Modge Podge on the letters because they are not coming off the lid, I'll have to ask....
On the inside of the lid she included directions for three different versions of the game. Super cute!