dc (GNU), 3 distinct characters (z|P), 453 bytes

Generator program: Try it online!

Because there can be a lot of possible different states of the stack, this solution prunes the states to be considered (only the topmost 2 can have value different from the default value pushed by z); therefore the result might not be optimal.

Analysis:

There are some options of using a digit, an arithmetic operator and P.

Possible cases for the arithmetic operator are:

• +: 1065 bytes. Throws a lot of errors. Generator program: Try it online!
• v: 7617 bytes. (you can get it from the generator program above by removing the solve(bestReprAdd), line in the generator)
• -: will probably result in the approximately-same or slightly-shorter program.
• %: obviously impossible. (because, for example, 11111 mod 111 is 11, so the constructible values are those already constructible with a literal or 0)
• ~: (unfortunately) also impossible because there's no way to discard the remainder without printing it out.
• / and |: should be possible. Analysis pending.
• * and ^: probably impossible, because there can only be a small set of representable numbers.
• i: An interesting possibility (11i to increase the input base, 1i to set the input base to 1); however the input base must be between 2 and 16, so there's no way to decrease the input base, so it's not very likely to work.

Other alternatives:

• z, P, (arithmetic operator): Terribly hard to analyze. Using zP| results in the solution above.
• z, P, o (since the output radix is not used, it's the same as pop the stack and do nothing in this challenge): 1601 bytes.