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Ryanair (and several other low-cost carriers) require that non-EU/EEA citizens present their passport for a visa check at the luggage counter:

Irrespective of a passenger's visa requirements, all non EU/EEA
citizens must have their travel documents checked and stamped at the
Ryanair Visa/Document Check Desk before going through airport
security.

However this seems to conflict with EU's Air Passenger Rights regulation, which only lists the following reasons for denying boarding:

Paragraph 1 shall apply on the condition that passengers:

(a) have a confirmed reservation on the flight concerned and,
except in the case of cancellation referred to in Article 5,
present themselves for check-in,

— as stipulated and at the time indicated in advance and
in writing (including by electronic means) by the air
carrier, the tour operator or an authorised travel agent,

So as long as I present myself to the gate (with a passport and valid visa) at the specified time, the airline is obligated to either allow me to board or cancel my booking and issue compensation as per the EU passenger rights?

As @Moo mentioned in his answer, I may also be denied boarding for:

reasons of health, safety or security, or inadequate travel documentation

But if I have a passport and a valid visa, is my travel documentation "inadequate"?

I had three flights on Ryanair during last two months. On the first flight I went to check in (checked in online, of course, but had time to kill), 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 3: @Zach Lipton found two cases when people were denied boarding by RyanAir. The first case was rather awkward as it seems like the guy did everything as requested (he mentioned checking in at airport and dropping bags) but still denied boarding. The second one looks legit, and it would be interesting to ask if they applied for, and get the denied boarding compensation. Unfortunately the story no longer exists on the original web site.

And mentioning the denied boarding compensation (and starting collecting evidence as soon as you are told to be denied) indeed could be game-changing.

Probably no, not by using that regulation, because the regulation has the following definition 2(j):

"denied boarding" means a refusal to carry passengers on a flight, although they have presented themselves for boarding under conditions laid down in Article 3(2), except where there are reasonable grounds to deny them boarding, such as reasons of health, safety or security, or inadequate travel documentation

As noted in the comments, by default you do not have adequate travel documentation until it has been verified as such - the airline is liable for your removal costs in the event that they land you and your are not eligible for entry into that country, hence the ability to deny you boarding for lack of adequate travel documentation.

The airline is always going to take the belt and braces approach in this instance, because the financial penalty is significant, and not allowing a refusal in such a case would mean the EU is basically forcing airlines to take ineligible passengers at their own cost.

I had the experience of one time not getting the stamp and was told it's okay this time but make sure to get one next time.

The thing is, it's really unnecessary to make someone go through an additional line for absolutely no reason. They do check it at the gate anyway. One time I went to get my stamp in Berlin, and the woman at the desk decided to leave her post and disappear for 10 minutes. The other lines were for checking in baggage, so I wasn't about to wait in those. I just got inconvenienced for a stamp that means pretty much nothing to anyone but Ryanair employees.