I'm afraid all three are agramaticales.
The correct sentence is
No me levanté hasta que el examen empezó.
"Hasta que" is a condition (a suspensive one) that sets a well defined point of time, hence imperfect hardly applies. The beginning of the exam is the trigger for a concrete action, so it demands indicative.
The only way to use imperfect there is in the sentence
No me hube levantado hasta que el examen había empezado
which seems to belong to Great Expectations or other literary work. And for the subjunctive case, the also written/literary
No me levantaría hasta que el examen empezara
No me habría de levantar hasta que el examen hubiera empezado
which tell about what would become the future at that moment of the story but now clearly belong to the past.
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