The Ultimate Question of Life, the Universe, and Everything is a complete unknown that has been debated by philosophers since the beginning of time. The answer, forty-two, was calculated by the supercomputer Deep Thought, but when that answer was revealed people were forced to realize that they didn't know what the actual question was. An even more powerful computer known as Earth was constructed to calculate the answer but was destroyed by a Vogon Constructor Fleet five minutes before the program's completion.
The Mice realized that the Question was probably encoded somehow in the Earthlings' brain-wave matrices, and made a failed attempt to find it by extracting Arthur Dent's brain. Marvin the Paranoid Android also once claimed to be able to see the Question in there but had never told anyone because they hadn't asked. However, as modern Earthlings are primarily descended from Golgafrinchans and not Earth's original inhabitants, the Question in their brains is probably not the right one anyway. The Mice decided to forego the whole mess and make one up.
Psychiatrists, led by the dastardly Gag Halfrunt, are greatly opposed to the discovery of the Question, as they rightly believe it will put them out of business.