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"spend just $1 and be rewarded in prizes that are worth exponentially more"

Any exponent of 1 is 1. So you wouldn't be able to win anything 'more' than that at all. Exponentially.

When people say a quantity like $x is exponentially increasing, they generally mean it's going as $x(1 + k)^t for some k, not as ($x)^t.

(And remember, any non-unitless quantity will be 1 in some units, nothing special about $1).

Perhaps taking powers of 1 reflects the futility of gambling? :)

Instead, consider the $1 to be 100 cents. (Do you think 40 degrees is 'twice as hot' as 20 degrees?) A matter of scale and perspective.