We did rent one of those rug doctors...that's what we cleaned it with initially.

That is the nature of the beast, it seems like it is perpetually expanding until the rug gets pulled out from under you.

In their case, they were moving to go back to graduate school and needed to replace the rugs from 13 years of cats. 

Obvious solution is to put down rugs but I'm hesitant as I don't want to take on that added expense.

If the tenants want carpet in their bedrooms they can get a rug.

I wonder if this is sustainable, and when will the rug be pulled out?   

To learn more about the prohibited transaction rules, see the following. https://www.irs.gov/retirement-plans/retirement-plan-investments-faqsIf you decide to invest your IRA in real estate, IRA Services Trust Company is in your area so it may be worth reaching out to them so that they can explain what their requirements are to hold IRAs for investing in real estate, as most big banks like Wells Fargo and Chase  don't offer IRAs that allow for investing in real estate.For the most part, the types of investments that cannot be held inside an IRA are:Artwork,Rugs,Antiques,Metals - with exceptions for certain kinds of bullion,Gems,Stamps,Coins - (but there are exceptions for certain coins),Alcoholic beverages, andCertain other tangible personal property.

The tenant could then lay area rugs and you would never have to install carpeting again.

Get some art, blankets, pillows, rugs, side tables, bookshelves (with books), etc. 

Could you  require or suggest the use of area rugs?