If left untreated, African trypanosomiasis will invariably kill the patient. The human immune system is unable to clear the infection.
I am aware of a few other infectious diseases with this property and I have a rudimentary understanding of the reason why. For example, I'm told that rabies has evolved to be uniquely good at attacking the brain because that's the only way it can spread to other hosts, and the neuroimmune system, being quite different from the peripheral immune system, is not capable of dealing with a virus that has this property.
I do not understand why the human immune system is ineffective against trypanosomiasis. One explanation I was offered is that the human immune system is just not very good at fighting parasitic infections (compared to viral and bacterial infections). However, this can't be the full answer, because there are other parasitic infections that may go away on their own without treatment.