Non-spherical (or non-circular) mirrors for reflecting telescopes are common and discussed in many places...
But what about elliptical, parabolic or hyperbolic lenses?
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Short answer: they exist, but they aren't cheap (unless made of molded plastic).
Long answer: Telescopes use parabolic, hyperbolic, elliptical, and less frequently, non-conic aspheres to achieve optimal performance for their design intent. A reflective telescope using 1 or 2 mirrors simply cannot achieve acceptable performance with spherical mirrors. Also worth noting, the traditional hand-grinding method of shaping a telescope mirror tends to form a parabolic shape anyway, so it just works out.
Transmissive optics (i.e. lenses) however, have the ability to put many elements/surfaces in a single axis, and thus can achieve better performance quality. Having said that, designs often limit the number of lenses for various reasons: size, weight, total transmission (since each lens absorbs some light), simplicity of assembly, and sometimes cost.
Thus, engineers will use aspheric surfaces to achieve these performance goals. In mass production, plastics and some glasses can be molded into aspheres with relatively low cost. Some can be formed by polishing a flat or spherical shape while the glass is under stress, and then when the stress is removed, it relaxes into an asphere. A lot of military/aerospace equipment use aspheres even if they are not low-cost, because it makes the equipment meet the size- and weight-to-performance goals.
There is even a common telescope design that implements an aspheric lens. The common Schmidt-Cassegrain telescope (SCT) has a front glass element which looks flat, but is actually a very slight 4th-order asphere ("w" shaped). This is manufactured commercially using the stressing method mentioned before.
What are the effects of using non-spherical lenses in refracting telescopes?
The largest refracting telescope is the Yerkes Observatory 40 inch aperture refractor and according to that article the lens with a 1.02 meter aperture and 19.2 meter focal length is f/18.8 with a mass of 225 kilograms.
If fine grained photographic emulsion was available in 1890 the telescope could have been made shorter by using aspheric lenses. However refractors are also kept long for other reasons. Even though the lens had achromatic correction, the residual chromatic distortion would likely have been worse for a faster (smaller f/no.) system.
Certainly for special cases where high resolution and wide field of view are both required at the same time, some complex lens designs with many elements will sometimes contain aspherical elements. But in these cases lens designers still prefer to tweak all of the spherical surfaces first rather than go for a more expensive aspheric surface.
The grinding process that makes telescope lenses and mirrors naturally tends towards spherical surfaces, making anything else is much harder, and doing it precisely is even harder.
But if you have to minimize the total number of surfaces for the objective, for example if you are limited to a single objective lens for mass reasons, then an aspherical lens based telescope is the way to go!
For an example of that, see this answer and Why does LRO's laser altimeter telescopes use lenses instead of mirrors?