A machine gives out five pennies for each nickel inserted into it. The machine also gives out five nickels for each penny.

Starting out with one penny, is it possible to use the machine in such a manner as to end up with an equal number of nickels and pennies?

As a bonus, is it possible to end up with exactly one dollar?

No, it is not possible to have an equal number of pennies and nickels, and it is not possible to end up with exactly one dollar.

You started with an odd number of coins, and every exchange leaves you with an odd number of coins. So the number of nickels can't equal the number of pennies.

And $1 in pennies and nickels is always an even number of coins. If the number of nickels is odd then the number of pennies must be odd in order to add up to $1. If the number of nickels is even then the number of pennies must be even in order to add up to $1. And since you always have an odd number of coins, $1 is not achievable.