Does anyone know a continuous group (not necessarily locally compact) with dense cyclic subgroup other than a torus? 



Dense cyclic subgroup
You already have some examples in the other answers. Groups which have a dense cyclic subgroup are called *Monothetic* groups. In the article "On monothetic groups" by P.R. Halmos and H. Samelson, you can find many of their properties, such as



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> Every compact connected separable (abelian) group is monothetic.
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