Absolute brain size is a poor predictor. Ratio of brain to body weight is better, and also ratios of executive brain area (e.g. more advanced parts) to the rest of the brain or the body.
So dolphins and humans would score higher, elephants not as much.
I think this is false. Our human intelligence, and other species that have intelligence, aren’t a path that evolution has followed. There’s no general trend toward more and more intelligence.
Human intelligence pays off because we use tools, because we have thumbs. It has to pay off big because intelligence is expensive.
A huge swath of the bookHow the Mind Works by Steven Pinker is dedicated to the OP’s question. His basic answer is: each incremental step had to pay off, and in some few lines it has. Other lines of evolution have eschewed intelligence (as we think of it) very successfully.
Well, that poster was talking specifically about human brain development.
One thing about humans is that our brain growth after birth is different from most mammals. Mammals have rapid brain growth during the fetal stage, but it slows down after birth. Humans continue the fetal rate for about a year after birth. It’s as if we are born a year premature, in that one area.
A large brain is benificial to almost any animal, but it comes with an expense. Its expensive energy wise to maintain a big brain. Therefore most animals develop a brain ‘just big enough’. Something like a mouse needs to know where to find food and how to avoid predators, but it does him little good to ponder his place in the universe. Instead of developing a big brain, he is better served by growing up and reproducing quickly.
Primates likely developed big brains because they were the first that were able to make good use of the intelligence a big brain could give them. Intelligence helped them work together in societies and create tools for example.
Dammit BUP. Beat to the post. 
With a true nuclear winter, you have plenty of frozen bodies to eat from. Our specie would survive.
WAG: Mammals developed larger brains than reptiles (by ratio to body size, of course) because warm-bloodedness requires more energy/food. When the dolphin’s and whale’s ancestors went back into the sea, they already had relatively large brains.
Assuming the radiation didn’t kill everyone.
True enough, not every line keeps growing in intelligence, or at least not at the same rate as the line that led to humans.
However, I always try to point out that the intelligence of our prehuman ancestors was growing long before we started hunting in organized ways, scavenging heavily, or doing a lot of tool use. It’s just not accurate to say that diet made our brains bigger. That ignores the fact that having opposable thumbs and precise, articulated, and strong fingers is the real correlating factor with big brains.
Close don’t count, except in horseshoes and hand grenades.
Indeed, the fact that we have come close many times, but have managed to avoid it each time argues FOR our intelligence as a species.
I’m not talking about situations where an intelligent species is too STUPID to avoid killing themselves off. I’m more talking about a situation where a cultural/language-based misunderstanding, and/or a failure of the communication or other technology involved, causes a nuclear or bio-weapon attack. Bad luck at the wrong moment, basically, is what I mean.
It’s almost certain to have happened many times, with the trillions of trillions of inhabitable planets out there. The question is, what percentage?
By what method are we going to be able to obtain evidence of even one extraterrestrial case of this? And without such evidence, how can this question be given a meaningful answer?
My brain hurts.
This planet has been home to probably more than 100 million animal species in its history (there are currently 8.7 million). One has developed the capability for civilization, and for nukes. Ever.
It’s not clear at all that getting a species that invents civilization is anything more than a fluke. Developing the capability for nuclear weapons is even less likely.
I would suggest that human intelligence is the product of several convenient coincidences.
As mentioned, we have grasping hands, and opposable thumbs. We can not only do crude lifting and carrying (without using our mouths) but we have the bipedal stance to allow it longer term - so we retain tools/weapons for a while and take them with us reather than relying on what’s at hand.
We have opposable thumbs and fine finger control that allows better grasping and manipulating. As a result, we can do fine work like scraping the pointy stick to make it more pointy.
We were relatively vulnerable when first venturing out into the meadows, so found a need for defensive weapons like club objects and pointed objects.
We hunted in groups and socialized in groups where interaction, “grooming” and cooperative hunnting also made big brains useful. (One theory is that speech evolved as a replacement for “grooming” as a status builder. You can only pick fleas off so many others in a day, but with gossip you can “service” multiple companions at once.)
Once we began to evolve the ability to plan and think, and then communicate the plan, and longer term memory it added a much larger possibility of success to endeavours like hunting. Even simple survival - if you can remember and share"we go down there because it was the safest place to cross the river in spring", then that is a step beyond wandering until you find a good ford or die trying.
So each of these conditions existed to some extent, but “smarter” individuals would be more likely to benefit. Like most of evolution, it’s the power of a positive feedback cycle.
But then you would have to deal with the Bunker Gap, which could determine the fate of our species.
I blame the monolith.
I agree.
My dog has enough brains and intelligence to smell out stuff (and perhaps to be able to form social bonds), thank you very kindly. It got him here, at least. Call that “small brain,” if you wish.
The humongous brains of other species must have similar satisfactory results. The brains’ hugeness is explained upthread as to the necessity for more processing, hence more meat. Why should size of brain meat have to do with that? Bats are small.
Two other things I’ve been thinking about:
In evolution, many organs or smaller systems just tag along, and have no role in the current stable position of the species. They didn’t get in the way of survival, and by the luck of the draw, they stayed put. Stephen Jay Gould has called them spandrels, a reference to the background space defined by the real architectural/design point of interest.
Can it be, then, that in the dolphins or whales, say, a large chunk of brain meat does nothing particularly important?
My second point is merely the Cartesian bias in many questions/answers–my own included here. Perception is not the be-all and end-all. Maybe percepts are. (Maybe goldfish dream.)
Evolution is a process of small changes providing small survival benefits, that add up through generations, leading to overall large changes. These small changes can interact in complicated ways, and thus not be a linear progression, and thus not be easily mapped after the fact.
I would say that what would lead a species to adapt intelligence like ours is a combination of mutations providing populations with more capabilities, an environment where the intelligence provides better means of survival, and a lack of hindrances from the intelligence increases.
Having humans around might be considered a hindrance.
We’re also “intelligent” enough to wipe out most other species - intentionally or unintentionally. Though we’re better at blanket destruction than selective culling.
The key for humanity was being in the right place at the right time with the right underlying features to start from.
Some semi-seriously propose this is why we’ve never been contacted by an extraterrestrial intelligence. But the reality is, we currently have no way to make any informed estimates for this. We have no knowledge of how cultures progress from developing nukes to complete world peace, or whatever, so we cannot even say how our one known example will succeed.
I suppose you can try to count up historical situations where, for a variety of reasons, we were on the tipping point and didn’t tip, but I’m not sure how you extrapolate that into some kind of statistic to apply to the presumed other intelligences “out there”.
I don’t think it’s a case of either/or, but rather each element contributed to the whole. A growing brain size combined with dextrous hands and upright stance, certainly was significant. But brains take energy, and meat is a much denser energy source.
One theory for why humans, specifically, have become so much smarter than other animals is the “Machiavelli theory”. The primary competitor and threat to humans, is other humans. If you are a lion, you only need to be smart enough to catch prey animals; if you are human, you need to be smart enough to deal with your fellow humans, who are on average just as smart as you are. So there’s been an arms race; if a human was born with genes that made them a bit smarter, then they had an edge in survival and reproduction; but unlike for example a lion that outwits gazelles, such a successful smart human breeds with his “prey”, and their descendents end up with the same genes and same advantages too.
Note that it doesn’t have to be a matter of violence either. A guy who has children because he talks a woman into sleeping with him or marrying him is just as successful in Darwinian terms as one who prospers by violence. More so, if he lives longer and produces more children.