I had three flights on Ryanair during last two months. On the first flight I went to check in, and after waiting in a long line the agent told me "you should have gone directly to the gate since you already checked in online". I asked her about the "required to" part, and she said, they don't care.
The next two flights I have skipped the check in and went directly to the boarding gate. When you go there, typically there is a line-up from the people who did the same. They process them in that line before letting you board. This happens without any hassle for both flights. I have also asked the gate agent, and she said there are always people - often from connecting flights - without those stamps, and they always process them at the counter. I asked her about the Internet stories about someone refusing boarding for that, but she said she never seen anything like that.
Also I respectfully disagree with Moo's answer regarding "inadequate travel documentation". Remember that your documentation is verified during the boarding - Ryanair doesn't simply take the boarding pass stamped "documents checked", they always insist on seeing your passport too, and they do look at it. Thus they have every chance to ensure you have an adequate documentation.
And of course if you are denied boarding, and bring your case to the arbitrage or the court for compensation, it would be easy to prove you have had proper travel documentation. The airline has different rules indeed, but the EU law is pretty clear you cannot be refused and denied boarding compensation because you did not follow the airline rules (unlike the US, where you can be denied compensation).
Update: I tried to Google stories of people who were denied boarding by not following the rules. I was unable to find any first-hand stories as well.
Update2: Reverted the edit back. Please don't edit the post simply because your spellchecker highlights "Ryanair" - that's the name of the airline, and it is spelled like that. It is not spelled "Ryan Air". Thanks.