How severe is the pain typically after having a discogram?

Fairly severe. Of course it depends on the individual but if you have a painful disc, the procedure can make it hurt worse for a little while. That's why I recommend that you have a plan in place beforehand to deal with the pain caused by the procedure. I recommend daily massage and opiate analgesics combined with an NSAID such as ibuprofen.
Painful. The pain of the discogram is limited to the test itself and usuall does not linger after completion, the pain is intense as it simulats an acute ruptured disc but for a breif time if the test is positive surgery usualy will fix the problem.