The short answer is there are 187 flavors of gum. This is calculated by the different chemical compounds that can be used in consumer bubble gum. Many scholars have painstakingly spent their careers and intellectual capital attempting to find a response to this question. This author is no different then those brave bubble gum flavor researching scientists. The search for the truth behind this question began online.
After years of filtering through bubble gum article after bubble gum related article only a little progress has been made. The tricky aspect of researching this topic is that nothing except the limits of imagination can hold the boundaries of bubble gum flavor. The vast array of cultures which have embraced the chewing tradition are numerous and varied. Some remote corner of the world may have dung beetle flavored gum and not have notified the rest of us. The point is that researching this topic was difficult and you should be thankful for these findings.
Answers from the online community:
187 - Maybe?
While the above answers are completely unhelpful, it gives some insight into the heated debate which rages around this topic. Digging into the truth to find how many bubble gum flavors exist will continue for eons into the future. To put it simply, the world may never know or even give a hoot.