American standard, erosion at bottom of toilet
I think @Daniel Griscom is right - that is urine scale building up.
A good overview here.

http://www.drain-clogged.com/urine-scale-in-a-toilet-remove-limescale/


  In general, urine scale in a toilet arises from a chemical reaction.
  Urine scale in a toilet forms when urine + water with high quantities
  of lime reacting chemically with one another.
  
  Once this reaction is in full progress it will lead to a
  yellow-brownish sediment in toilets or urinals – usually in a rather
  solid form. The lime contained in the toilet water forms a chemical
  bond with the uric acid and the urine, more precisely a lime and
  magnesium compound. The solid substances can then be recognized as
  so-called urine scale sediments. Furthermore, bacteria play a role in
  the formation of urine scale in a toilet. The longer urine scale
  remains inside the toilet, the more difficult it will be to dissolve
  and remove it.


It is a combination of hard water, urine and germs.  I think you need all 3.  Interestingly I think some people with chronic urinary tract infections can get bladder stones (staghorn calculi) by the same mechanism.  

Acidification should dissolve the buildup, very eventually.  Various sites recommend turning off the water line, flushing out the water then leaving something acid to sit on the scale.  You could use vinegar but I am also American, and impatient, and if I am going to use acid I want something that packs more punch than salad dressing.  You could get an acidic cleaning product and lay that on there.  If I were doing this I would start with Bar Keepers Friend

https://en.wikipedia.org/wiki/Bar_Keepers_Friend

which has oxalic acid in it and is OK for porcelain.  Plus it is a powder so it will be easy to get it to rest on top of your exposed deposits.  It will take more than one application.  Or you could get some heavy duty acid drain cleaner (look at the label!) and try that.  

Or you could scrub it away / chip it away.  That stuff is hard and removing it mechanically will take time.  



Then you have to keep it from growing back.  If you don't want to flush you could add one of those products to the tank that suppresses microbial growth in the water.  There are lots of them.