Can I have an iTunes library on a hard drive connected only to my wi-fi router? I have both Windows and Mac laptop computers with hard drives that are too small to fit my music collection or backup iPhones.  I would like to purchase a hard drive that I can connect to my wi-fi router which can include parts of my iTunes library not on my laptop, and could be a place for me to backup my phone.  For example I was thinking about the Seagate Central 3 TB Shared Storage Ethernet External Hard Drive.  Does anyone know of a way to do what I want with this or any other product?  Thanks.
Yes, you can store your iTunes library on shared drive. Its not really any different than an external drive attached via USB or the like.

Apple KB: iTunes for Mac: Moving your iTunes Media folder

If you are looking to run home sharing, or stream to an AppleTV or the like you still need a computer to handle that.