Interesting... google crawls something like a few petabytes of data a day, I believe, they crawl the whole internet in one week, so we can assume that the internet has around 2e16 bytes of data in it(images included) so knowing that a large amount of that will be images, lets assume that it's 50% images(images take up more space than text, but there are fewer of them, presumably) so it's 1e16 characters, and 1e16 bytes of images. Most images are in jpeg format, a 3000x2000 image in jpeg format takes up about 2e6 bytes, so that's 5e9 3000x2000 images(this is probably an underestimation, jpeg is one of the more compressed formats, I'll underestimate later) so if 2 of those pictures take up about the same area as a 11x17 sheet of paper(that's A4, I believe), that's about 2e9 pages of images(underestimate on the division). Now, add that to the 1e16 charecters, since an 8.5x11 page can hold approximately 4e2 words, that's 2e3 letters, an A4 page could hold 4e3 letters, so that's 2.5e14 pages of A4 paper, total so far we've got just about 3e14 pages of A4, but this is SciOly, we're going to put 4 standard pages to a sheet, so now we're right around 7.5e13, this seems a little high(on pages of text, honestly) so I'll just round down and say 5e13 pages. Since a page is 2e-3 inches thick, that's 2.5e10 inches thick, since you didn't specify units, I'll answer in inches and put down 10.
So the internet contains something like 46 billion web pages(http://www.worldwidewebsize.com/
), 5e9 pages, using this site (http://httparchive.org/interesting.php
) we can figure that 7.5e5 bytes of images(I'll focus on images first) are going to be requested per page. That's 3.75e15 bytes of images in the entire web. Now, there's on average 5.3e4 bytes of HTML per page, since there's not stats on how much of this is visible text, I'll estimate that 30% of html is actually visible text(disclaimer: I do web development for the company I work for and admin our SciOly site). This translates to 8e13 bytes of text on the entire internet. By opening up word and selecting size 8pt font(what scioly sheets seem to use) I know that 1e4 characters can fit on a 8.5x11 sheet of paper, so about the same on A4(which, it turns out, is 8.25x11.75, whoops). Since a byte of ascii is a character, that's 8e13 chars in the internet that would actually print, since I assume this is double sided, that's 4e9 pages of text. Now the images are much harder... taking the weighted average of the image sizes(from the HTTP Archive) we can find that every picture on the internet would take up 1.44e17 pixles, average printer dpi is 600, so that's 3.6e5 pixels per square inch, thus 4e11 square inches of images on the internet. A sheet of A4 paper is 9.69e1 square inches per side, so 2e9 pages of images. Total, we get 6e9 pages of information on the internet, since a sheet of paper is .003" (I measured it), that's 1.8e7 inches thick. Thus a fermi answer of 7.
Wow that was difficult... Here's an easier one from a recent test I took. If one planted grape vines 5 feet apart, how many grape vines would cover central park?