Xpendable:
Seriously? You're recommending a more expensive board for something that requires at most 4 digital inputs (2 user inputs, 1 coin indicator input, and 1 opto-interrupter for the carousel), 2 motor output signals, and a single 7-seg display... The game logic is so simple that it would probably take 20 lines of code... maybe 100 lines by an idiot. I have trouble seeing how a complete newbie would even come close to hitting the SRAM limit or the program space limit. The original firmware of my MonsterShield (which is about 1000 times more advanced than this thing) was close to maxing the memory, and that was a LOT of code. Not saying that that there aren't times when you need more... but for THIS project? Seriously?
Your argument is based on experience, so I will concede to that. The 328P is a great workhorse, and certainly has the I/O to handle the Ops needs. It does seem prudent (in retrospect) to start with the UNO or variant and push that to the limit before jumping to the larger uC.
Thanks for the reality lesson.
Ray