What's the risk of contracting HIV from a half or quarter syringe of filler?

The practice of using any filler other than your own syringe and almost 100% of the time used at a single injection setting is highly unusual.  I would really question your choice of injectors. 

Saving "your"  previously used syringe could open the process to very human mistakes if stored with 100's of other patient syringes,i.e.,misreading, similar names, loss of label,etc. Filling a new syringe from a master source syringe that has never been used to inject anyone is safe.

Some of the dermal fillers come in 0.5cc syringes that can be used for touch ups. Unfortunately, these are not half the price. Once a syringe is open, it should not be reused. There really is no cheap way to safely do fillers. There is alway some natural asymmetry so it is unrealistic to expect result to always be perfectly even. If it is not obvious, my first advice is to just leave it alone. If you and the doctor really feel you need more filler, maybe the doctor can work with you on the price of the additional syringe, but don't take chances. it's just not worth it!

When you purchase filler, it should be by the syringe, if all is not used, it should be labeled as yours and placed in the medical refrigerator.  The notion of dividing a syringe among patients is improper.  

it is highly unlikely you would contract HIV but this is not safe needle practice /precautions and not recommended  by CDC. Google cdc injection safety 

I would think there are no studies that were done on that subject and unlikely there will be any. I would be very cautious. Many times the aspirations are done during injections to avoid entering a vessel and patients blood can enter the syringe. Even if the needle is changed immediately before syringe is stored, it is not safe to use it on a different patient. If doctor saves syringe after injecting you specifically and you want to save it for the future, then he/she should change the needle before storage and it may be fine. SAFETY FIRST!

There are multiple problems with using previously used or saved filler syringes. First, any syringe that is saved can be inadvertently contaminated and cause an infection or other problem when finally injected. It is also easy to mix up syringes between patients accidentally. A syringe should NEVER be injected in more than one patient, even if the needle is changed, because of the almost 100% certainty that some fluid or blood from the first patient will enter the syringe. Although the risk of transferring HIV or Hepatitis is not great, it is certainly possible. As no reputable injector does either, if your Dermatologist does as you indicated, I would suggest you have someone else do your injections. The slight savings in initial cost is not worth the possible long term risks and their significant costs.