Step 1: Recall which diseases mosquitoes carry.
Mosquitoes are vectors, meaning they carry disease causing organisms from one host to another through their bite. Several well known human diseases spread only through mosquito bites.
Step 2: Confirm the three mosquito-borne options.
Dengue fever is caused by the dengue virus and is spread by the Aedes mosquito.
Malaria is caused by Plasmodium parasites and is spread by the female Anopheles mosquito.
Filariasis is caused by thread-like worms such as Wuchereria bancrofti and is spread by the Culex mosquito.
All three of these therefore need a mosquito bite to pass from person to person.
Step 3: Examine sleeping sickness.
Sleeping sickness, also called African trypanosomiasis, is caused by a protozoan parasite of the genus Trypanosoma, not by a virus or a worm. This organism is carried by the tsetse fly, a completely different insect from the mosquito, which bites during the day and lives in parts of Africa. The parasite enters the blood and later the nervous system through the tsetse fly's bite, never through a mosquito.
Step 4: Final conclusion.
Since sleeping sickness travels through the tsetse fly and involves a protozoan parasite rather than a mosquito, it is the one disease among the four that is not mosquito-borne.
\[ \boxed{\text{Sleeping sickness}} \]