Suppose you were required to play a zero-sum game, but could choose whatever probabilities and payouts you wanted, so long as the expected cash value came out to zero. (If necessary, assume "the banker" has infinite resources.)
It's a loaded question because people play games for the game content as much as (more than?) the payout (especially if the EV is 0!). For example, you could choose up to eight rounds of reverse Martingaling -- start by betting $1 even money; if you lose, you finish down $1; if you win, you double your bet. 255/256 chance of -1, 1/256 chance of +255. The fun there is in the sequence more than the exact odds. Despite all that, it might be simpler to focus on odds/payout until you find a fantastic game that simulates those.
Mind, a lot of people are so cash risk-averse that their answer would be "play some fun game with play money and guarantee a cash outcome of exactly zero." That's fine too.
An interesting related question: How would your answer change if the expect cash value had to be a loss of one dollar? Would it just be the same answer as before, with a $1 surcharge tacked on all outcomes, or would it be different?
If the payout has to be zero and the game structure is ceteris parabus, then I think I'd require three things from the ideal version of all this:
1. A tiny chance of a huge payoff
2. A cap on my losses
3. A greater likelihood of winning than losing
Given all that, a quick and dirty approximation that works out correctly: Consider one million possible outcomes (for example, six lottery popcorn machines, each with the ten digit balls).
BIG MONEY: One in a million apiece for $256K, $128K, $64K, $32K, $16K, $8K, $4K, $2K. That's eight outcomes down, $510K. Add 40 more ways to win $1K, so we're at 48 outcomes $550K. Add two break-evens to make the math rounder.
ALMOST BIG MONEY: Add 500 payouts of $100 each. Now we're at 550 outcomes, $600K.
SMALL MONEY: Add 10,000 payouts of $10 each. Now we're at 10,550 outcomes, $700K.
MORE ROUNDING: Add 89,450 break-evens so we're at 100K outcomes, $700K. We need 900K outcomes totaling -$700K, which is an even multiple of nine outcomes totaling -7.
THE MEAT OF THE GAME: 600K outcomes of +$1, 200K outcomes of -$4, 100K outcomes of -$5.
Revisiting my requirements above:
1. More than a 1 in 100K chance of winning at least a thousand, in fact a shot at 256K!
2. Maximum possible loss per game of $5.
3. 60% chance of winning exactly one dollar.