Chewing gum has many practical uses; it's a good idea to always have some on hand.
The most obvious use (and perhaps the most common) is to guarantee one has good breath. Any gum with a strong mint flavor will give one's mouth a minty scent strong enough to mask most unpleasant breath odors. Unlike breath mints, which can only be in the mouth for a few minutes after which time any breath freshening effects wear off, gum can be chewed for hours with breath freshening benefits lasting about as long as the flavor.
Once thoroughly chewed, gum is very useful for the occasional objects that might need stuck together or for filling in small holes. While extremely sticky, it isn't particularly strong and shouldn't be used for anything requiring more strength than clear office tape.
One can anonymously get revenge upon one's enemies with gum. One tactic is to stick the gum somewhere they are likely to stick one of their body parts in a short amount of time (it's important that the gum doesn't dry before they touch it); doorknobs, the undersides of desks and coat pockets are all fair game provided one could avoid being caught. A particularly cruel trick is to stick gum to their hair; while much riskier it is quite effective especially if said enemy has long hair.
Gum is somewhat effective in avoiding conversation with others. While chewing one's mouth appears somewhat busy, which combined with another task such as reading can keep others from provoking small talk. For a greater effect, chewing gum loudly with one's mouth open and blowing large bubbles that pop all over one's face further deter unwanted social interaction (although it could be argued if one has bad breath not chewing gum at all would be more effective).
Because chewing gum is similar to eating, it can help suppress one's appetite, which is particularly useful when dieting. Even in sugary gum, the calories consumed are negligible and made up for many times over by the reduction in appetite.
In many settings having a pack of gum can act as somewhat of an icebreaker. While perhaps not as obligatory as donating a cigarette, it is considered polite to give a piece to someone who asks provided there is more than one piece left in the pack. The gum can then be used as an excuse to start small talk; starting out with generosity aids in making a good first impression.
There are many gums that are made for a specific purpose, more commonly known gums are nicotine gum for smoking cessation and dental gum to clean and whiten teeth. Powerful caffeinated gum is used by the military where sleep deprivation is common and brewing/drinking coffee can be impractical. Less powerful energy gums are available to civilians as well.
Lastly, one cannot forget how useful gum is for retrieving small valuables that have fallen down drains.