I did some article browsing and here's what I found:
- Almost all of the articles I came across discussed the affect of general anesthesia on certain nerve tissues, ion channels, and receptors. I could not find a good overview or any peer reviewed articles covering the long-term clinical effects (what impact it actually has on the patient) of general anesthesia.
- Many of the studies are either in animals, in children, or in the elderly; not general population or a wide patient base.
- A general consensus seemed to be that the possible short term effects from anesthesia (malignant hyperthermia, hypotension, slowed respiratory rate) and the surgery itself lead to many more complications than any long term effects from general anesthesia.
I did stumble across one fairly broad article which discussed cognitive decline in the elderly after uncomplicated general anesthesia. The syndrome is called Post-operative Cognitive Dysfunction (POCD). The article didn't seem to consider how many times patients had undergone anesthesia in their lives, but focused more on receiving anesthesia at an old age (85 or older).
Lewis, M. C., and I. Nevo. "Uncomplicated General Anesthesia in the Elderly Results in Cognitive Decline: Does Cognitive Decline Predict Morbidity and Mortality?" Medical Hypothesis 68.3 (2007): 484-92. Science Direct. Web. 19 Aug. 2016.
Here's another article that doesn't really answer your original question, but might be helpful nonetheless:
General discussion TIME article
So I couldn't find any articles addressing your specific situation, but hopefully some of this info can help you draw your own conclusions, and maybe someone else will have better luck article searching than I did.