A little thought experiment that will help lay the groundwork for later posts:
A rational being is one which always acts to maximize the likelihood of producing optimal outcomes. In other words: A rational being has some set of pre-rational preferences, and a body of available information. Using this information, the rational being evaluates the available courses of action and chooses the one that appears likeliest to produce the most-preferred (optimal) outcome.
We thus already see that a rational being cannot exist without something irrational, namely preferences. But there's a deeper way in which irrationality can be rational.
Imagine that you are in the jungle. You, like all human beings, have a capacity called agency attribution. This is, effectively, an alarm system that notifies you when a sensory input is caused by an intentional act. In other words, it's what distinguishes between leaves rustled by the wind (non-intentional), and leaves rustled by a hungry tiger stalking you (intentional). If you were perfectly rational, you would assign agency wherever there was evidence of an intentional being, and no other time.
However, here in the jungle, tigers are pretty good at hiding. There's a good likelihood that those rustling leaves are a tiger, even though there's no evidence that it's anything other than the wind. If you fearfully and irrationally shoot at every rustling leaf, on the knee-jerk assumption that it's a tiger, you are likelier to survive than if you're purely rational about it.
Increasing your survival chances is, generally, a pretty rational thing to do. So, in this circumstance, behaving irrationally on one level is actually rational on a higher level (meta-rational?) Indeed, I'd speculate that something like this scenario creates a selection pressure for a hair-trigger agency attribution, which would explain why it's a nigh-universal human trait.
Anyway, if this scenario seems oddly familiar, it should be: it's Pascal's Wager, only with a tiger instead of a vengeful God. I'm not arguing that Pascal's Wager should be regarded as a valid argument for belief in God (the difference is that tigers can be demonstrated to exist); only that the kind of irrationality which leads to Pascal's Wager serves a useful function.
Edit for Clarity: What I am arguing, in essence, is that there are circumstances under which a hypothetical, fully rational being, who prefers survival to death, will wish to be less rational. To put this yet another way: there exist selection pressures favoring certain forms of irrationality.