Is it possible to make caramel that does not taste sweet?
It is very simple, you just have to heat it long enough. It can even happen by accident :)

The taste is a mixture of bitter and sour, while the smell component is mostly towards something burnt. 



Also, your assumption "because of the uncaramelized sugar mixed in the caramel" is incorrect, or at least incomplete. There usually is such sugar, but many of the caramelization products of the early stages of caramelization also taste sweet on their own. In later stages, you get more and more nonsweet stuff until you get into the state I described (barely liquid charcoal).