Who offers cheap liability insurance?

It depends on what kind of liability insurance you're seeking. If you want car insurance, Geico and StateFarm are generally very inexpensive, though it all depends on your car and driving history. If you want business liability, look at TechInsurance or Nationwide for the best fit.

If you are an artist, you can try ACT Insurance Program - cheap and simple artist liability insurance, starts from $265 annually or from $39 per show.

Also there are companies that sells product and general liability insurance for businesses and individuals.

By no means should you shop by price only. The amount of premium that you are charged depends upon a number of factors, not the least of which are policy limits (the mount of insurance that you buy) --liability limits.