Why is intelligence so rare?

This is just a question:

Whether or not life itself is a miracle, or the product of divine action, on the planet we live on life is all over the place. Life has existed on Earth for billions of years. Living creatures inhabit all sorts of environments, some of them seeming almost completely uninhabitable.  

And yet, so far as we know, no species except our own has attained our level of intelligence in all those aeons. Why is this? 

Our own history tells us that intelligence is an extremely useful trait to acquire; without it we would certainly not be kings and queens of the jungle. That suggests that if this trait had evolved before, it would have been likely to flourish. 

Is the reason it didn't evolve before that the ability to think as we do is such a huge genetic alteration that its arrival could require billions of years to take place as a matter of sheer biological luck? Well, the evidence doesn't seem to support this "huge alteration" idea, because lots of creatures have some measure of cognitive ability. At least some other animals, including parrots, dogs and chimpanzees, can learn something of our languages (which seem to be one of the most distinct of our intellectual accomplishments). Any number of animals -- perhaps even plants! -- have enough cognitive capacity to form purposes and try to achieve them. Our intelligence is greater than all theirs, we believe, but it doesn't seem totally unlike what they have. 

So if life on earth is prolific, and intelligence a useful and genetically reasonably available trait, and if evolution over millions of years has explored one way after another for achieving survival, why was intelligence so late to show up?