What are the benefits of a large refracting telescope? Currently, refractor telescopes are limited in aperture because gravity will deform the main lens over time. According to Wikipedia the largest practical refracting telescope is 1m. 

If we were to overcome this limitation, what would be the theoretical benefit of such a telescope that other types of telescopes currently can't give us?
This is a really interesting question!
The Dragonfly telescope described below takes advantage of the fact that on Earth, the optical surfaces of some lenses can generate less diffuse scattering than the optical surfaces of reflecting telescopes. If you were worrying about scattering from secondary mirrors you could make an off-axis primary, but at a nanometer scale a metalized mirror will have more roughness than a glass lens with antireflection coatings (or so the links say). I suppose there are broadband dielectric reflective coatings for telescope mirrors, but that's probably a topic for a new question.
note that in many cases surface brightness limitations are often the result of  skyglow (artificial) and airglow (natural) rather than telescope optics, so applications for large lenses might be found above the atmosphere.
If we ignore the fact that in the solar system space is filled with micrometeorites, then a lens in space might maintain it's lower diffuse scattering for quite a long time! But we can't, so it may not. It may get beaten up fairly quickly in fact. A long tube in front of the lens will decrease the solid angle of primary impacts, but impacts inside the tube may still cause trouble, and if you don't shade the tube from sunlight, diffuse scattering from the close-to-but-not-completely-black coating might also cause trouble.
So far there are no answers to How fast do optical surfaces get dirty or damaged in space? that I can point you to.

Below's discussion of and links to the Dragonfly Telescope is taken from the question What (actually) is the “deprojected half-light radius” of this almost-all-dark-matter Galaxy?

The recent news of the Ultra Diffuse Galaxy (UDG) Dragonfly 44 is an excellent example of what could be termed 'observe different' thinking. The dragonfly telescope is noted not for the size of its collective aperture, but for the absence of the diffracting effects of secondary mirrors and surface roughness that limit the contrast of dim objects in conventional telescopes when brighter sources are nearby. See here and here and here.

above: image of a Dragonfly refractive array telescope from here. Image: P. Van Dokkum; R. Abraham; J. Brodie

above: The Dragonfly 44 ultradiffuse galaxy from here. &quot;Dragonfly 44 is very faint for its mass and consists almost entirely of dark matter. (Pieter van Dokkum, Roberto Abraham, Gemini Observatory/AURA)&quot;