If there were only one cold virus, it would make you sick for a week, and that would be that. After you recovered, you’d be immune from colds (though perhaps that immunity wanes with time).
Unfortunately, there are lots of cold variants, so we keep getting sick with new ones. Suppose, however, that you got sick with (say) 10 different cold viruses at once. Would your symptoms be much worse than if you were sick with only one? Could this be a way to get immunity to colds?
Hm. Good question. The best case scenario is that you will become immune to the 10 strains of the cold virus that you encountered–there isn’t likely to be anything about having multiple strains at once that would lead to stumbling into a universal cure. The worst case scenario…off the top of my head, I don’t think that it would be worse than having one strain–the immune system is “massively parallel”, no reason that it shouldn’t be able to work on immunity to several viruses at once. In fact, many modern vaccines work on that very principle, mixing vaccines for several conditions in the same shot.
I suspect it would be the same in terms of illness as being infected with one strain for a few reasons:
The different strains of virus will be competing with each other to infect your cells so it’s likely that only the most infectious one will make any decent headway
The ill feeling from being infected with a cold virus is mostly your own immune response, this isn’t going to ramp up 10x because there’s more types of virus in your nose/URT
even with a single virus strain, your immune system will develop polyclonal antibodies Polyclonal antibodies - Wikipedia against it (several different types targeting different bits of the virus) and each antibody is produced by a different cell lineage so they shouldn’t interfere with each other…you might use slightly more energy if there’s 10x more antigens to develop antibodies to but I doubt it would be significant.
The symptoms you feel - fever, stuffy nose, headache, fatigue - none of them are caused by the virus itself. It never gets a chance to do enough damage to matter, at least for illnesses that humans survive. So with 10 viruses those symptoms are going to be worse and longer lasting, I think. But only a little bit worse - not 10x worse. You might have a slightly higher fever, slightly worse other symptoms.
2.5 years ago I posted a set of related questions that didn’t get any answers. In acknowledgment of Cunningham’s Law*, maybe by responding to your question I can induce action in that older thread. 
So: my anecdotal experience is that a specific cold virus has a characteristic spectrum of symptoms with somewhat characteristic severity, as observed when the same virus hits multiple people at the office or in the household. If this is true – i.e., if there is a correlation between symptom type/severity and viral strain – then getting multiple viruses at the same time would compound things at some level.
(And if this isn’t true, I’m still very interested in informed replies to my older thread where I ask about such correlations.)
[sub]* “The best way to get the right answer on the internet is not to ask a question; it’s to post the wrong answer.”[/sub]
Well, immune until that cold virus mutated. And since a ‘generation’ for such a virus is about 150 hours, they mutate at a pretty fast rate compared to humans. Fast enough that there are new strains every year. Sometimes even new strains spread between the spring when doctors decide which strain(s) to put in the flu vaccine, and the fall/winter flu season.
Would having 10 cold viruses floating around the body up the chance of gene transfer between them to create a new virus? <ready to learn>
It’s possible. Keep in mind that this probability is very low. One problem with this gene transfer is that viruses need to be very small, there are numerous pressures on them that limit their size. And a gene borrowed from another virus may be incompatible, in the same way that you can’t make a better Toyota Yaris by borrowing parts from an F-150.
But you can, say, shave a little metal from a Yaris part and the car will probably still work. If the change actually makes the car better, you could slowly make a series of small changes resulting in a vehicle “evolving” into something different.
Viruses can mutate all on their own - they usually don’t have any error correction genes, and so when they get produced by the victim cell, errors in the copying process are common.