Is it possible to become addicted to sleep deprivation?
Assuming there's not a neurological dysfunction underlying sleep deprivation (which is even more possible with Aspergers as sleep dysfunction is a typical comorbidity) it can simply be a learned behavior.  The more you do something (whether you particularly "enjoy" it or not) the more likely you are to build it up as a habit.  Procedural memory is always at work.

So in this regard, it can be addictive behavior in the same way a "bad habit" can.

I wouldn't call it a drug.  Pharmaceutical drugs are defined as the intentional delivery of a substance from externally to inside the body: most commonly acting as an allosteric modulator on neural receptors.

About 73% of children with Asperger's experience problems with sleep according to the Asperger's Association of New England.