I've read that a human body contains about 150,000-180,000 km (Pakkenberg et al., 1997; 2003) of nerve wiring in the whole body. But does this length increase with the size of the animal? For example, an elephant is much bigger than a human and perhaps even his brain is bigger but is an elephant also the animal with the longest nerve cells in total?
There is no reason to believe that total nerve "wiring" (or combined length) does not increase with the size of the animal, although the number of neurons in the brain (and the number of connections they make via dendrites) matters as well.
Animals with long necks, legs and tails have long nerves. Ancient sauropods may have had the longest nerves (referring not only to nerve bundles but single neurons) in history. Their brains were not as large compared to their body size, however, as humans.
But whales - the largest animals on earth - do have larger brains. Modern cetacean brains are among the largest in both absolute size and in relation to body size of all mammals. The brain of an average adult sperm whale weighs 8 Kg; by comparison, the average adult human brain weighs about 1.3 Kg.
Blue whales are the largest animals in existence today. With a large brain (3.6 Kg) and a long body, I can't see a particular reason to believe that the total amount of "wiring" in a blue whale doesn't exceed that present in a human.
[Normal weight of the brain in adults in relation to age, sex, body height and weight]
Normal organ weights in men: part II-the brain, lungs, liver, spleen, and kidneys.
Cortical complexity in cetacean brains