Final answer:
After becoming a white dwarf, a star's next step is to cool down and fade away entirely, potentially becoming a black dwarf given enough time, although none are thought to exist yet due to the universe's age.
Step-by-step explanation:
After a star has evolved through its main-sequence phase and has become a white dwarf, its future evolution is quite predictable. The star at this stage has ceased nuclear fusion and is supported against further collapse by electron degeneracy pressure. Eventually, the white dwarf cools and dims over a very long time.
It is theorized that it will ultimately become a black dwarf, an extremely cool object that emits virtually no heat or light. However, if a white dwarf is in a binary system and acquires enough material from its companion star, it may surpass the Chandrasekhar limit and ignite a Type Ia supernova.
Considering the vast time scales involved, the transition of a white dwarf into a black dwarf is purely hypothetical, as the age of the universe is not yet sufficient for any to have formed.
In summary, a star's evolution after the white dwarf stage, if it remains below the Chandrasekhar limit and is not part of a binary system where it could gain mass, leads to its eventual cooling into a black dwarf, which is the final stage of a low to medium mass star like our Sun.