Does anyone know if there is a name for the curve which is a helix, which itself has a helical axis? I tried to draw what I mean:
According to this journal article, we may call it a doubly-twisted helix or generally a multiply-twisted helix.
For a light bulb the wire is called a "coiled coil filament" in this Wikipedia article.
The German word is "Doppelwendel" (roughly translated: double screw).
I don't believe there is a standard term. Nevertheless, MathWorld calls the resulting curve a "slinky".
Of course, one can have a "slinky" of a "slinky":
A corresponding construction on the torus is called an "iterated torus knot", so perhaps "iterated helix" would be a good name. I don't know of any standard name.
Coiled coil as others also stated. More length is packed in a small volume. Used for electric bulb filaments, toroidal transformer primaries etc. Polar orbits of sun synchronous satellites around earth/sun is another example if the inner coil has no torsion.
A constant vector component of binormal is added to central coil vector if helicoids are to be defined as a surface... as a set of connected coils.