Nyquil - will daily use for more than 2 to 3 weeks get me addicted?

As the others have said Nyquil is not a good choice for sleep. It contains almost as much alcohol as a shot of whisky for one thing. It also contains a strong decongestant and antihistamine which you dont need unless you have a bad cold. The antihistamine in Nyquil is available OTC in a sleep preparation without all the other ingredients that you dont need. It is called Unisom. This has doxylamine and is the ingredient that helps you sleep. Here is a link with info on this medicine.
https://www.drugs.com/mtm/unisom.html
You might want to discuss your inability to sleep with your Dr however. Insomnia can be caused by many health problems but if it is just plain insomnia there are many good drugs that are availble by prescription that are very effective for insomnia.

I don't know the specific ingredients of Nyquil, but as Suzzane answered it is not the answer to your sleep problem. It does contain alcohol for sure, & in combination with the cold/allergy med in it will make you sleep better. Try taking a couple of just benadryl & see if that alone helps you sleep. If not, then I suggest you speak with your doctor. It can really throw your whole system off if you are not getting a good night rest. Just my thoughts ...

Nyquil Cold medicine is not intended to be used solely for sleep purposes. There are safer products available that you can take on a more regular basis. You should talk to your doctor.