In informal text or speech, you can use "and" many times in a list. This often sounds rushed or childish, though. Hot Licks' example of "Lions and tigers and bears, oh my!" is from a movie scene with a scared child.
In formal writing, it is normal to have one "and" for each list or sub-list. The original post has a main list, and a sub-list. Both versions of the original list are acceptable in formal writing. The second version is less ambiguous.
- "I am a good person, an exemplary citizen, and I want to make the system work better."
The first version can be parsed as either
- a list containing three items (good person, exemplary citizen, I want…), or
- a list containing two items (I am a good person, I want…), plus an appositive phrase (which is similar to a parenthetical note).
- "I am a good person and an exemplary citizen, and I want to make the system work better."
The second verse can be parsed as
- a sub-list containing two items (good person, exemplary citizen), and
- a main list containing two items (I am <the sub-list>, I want…).