According to the WHO:
67% of the population are infected with herpes simplex virus type 1 (HSV-1)
Presumably there are other viruses as well that infect a lot of people. How many infections does the average person have?
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Ignoring short-term (acute) viral infections, so only considering persistent viruses, probably on the order of 5-10-ish. There are 8 human herpesviruses known, of which 6 (herpes simplex I, EBV, CMV, VZV, HHV6, and HHV7) are very common in humans, generally infecting 50-90% of a population. There are over 50 strains of human adenoviruses, but only a few are very common and persistent; hAb2 and hAd5 are probably the most likely. Then there may be a human papilloma virus or two persistently infecting many people, though the prevalence is lower than herpes or adenoviruses. So, it's reasonable to assume that most people have somewhere around 5-10 persistent viral infections.
If you add in acute infections, and ask about all exposures during one's life, you're probably looking at thousands or tens of thousands of viruses, but the questions about what's a different virus and what's an "infection" become trickier.