This is what Brazilian neuroscientist, Suzana Herculano-Houzel, aimed to find out. The African elephant brain is three times larger than ours, but did it have more neurons? Over a period of six months, she and her students sliced an elephant brain by hand, separated out the different brain structures, and processed them into 5 gram pieces. The pieces of brain were stained and the neurons counted by hand.
The African elephant brain had three times the number of neurons in a human brain, 257 billion to our 86 billion. But, 98% of their neurons were located in the cerebellum, in the back of the brain. That left 5.6 billion neurons in the cerebral cortex of the elephant compared to 16 billion neurons in the human cortex.
The answer was that the human cerebral cortex has almost three times as many neurons as the elephant brain.
"So what do we have that no other animal has? A remarkable number of neurons in the cerebral cortex, the largest around, attainable by no other species, I say. And what do we do that absolutely no other animal does, and which I believe allowed us to amass that remarkable number of neurons in the first place? We cook our food."
Here's a link to the article that she wrote for Nautilus. It's a fascinating read.