I recently had to print an entire website, with about 20 pages, and the good thing before I started visiting each page and hitting print is that I looked at the Options tab and I've seen that you can easily print an entire website from Internet Explorer.
1. After you open the website you want to print, go to File and select the Print option.
2. You have to select the printer you want to use and set the options for that before printing. For this test I used a virtual printer driver (PDF maker).
3. Go to the Options tab and check the "Print all linked documents" option to print all linked pages from a website. Select Print in the lower left corner and it will start printing the pages in the order it reads them.
Now, the problems that could appear.
The main problem is that there could result an infinite loop. If a page links to another page that links back to the original page, the print job might loop continuously. Also, if the file has outside links then you could end up printing external pages that are not needed.
A solution to these problems is to first use the Print table of links option (and with the Print linked documents option unchecked). With this option checked, you will print a page that contains all the linked documents in that website. You could verify which links are printed, and if everything is ok you can do the Print linked documents option.
Another caution measure is to use the Print to file option (from the general tab), that will print only "electronically" the files.
I've tested this with Internet Explorer 6 and Maxthon. I know for sure that this option is missing in Opera, and I'm not sure if FireFox has it or not.
P.S. Happy new year everybody!