While discussing the BRRRR Strategy on page 70 of The Book on Rental Property Investing, the author says "Sure, at some point, the bank will stop refinancing the properties for you."
Why is that? Is the strategy frowned upon by banks?
It depends on the lender, but in general, no, it's not frowned upon at all.
It mostly depends on the bank's comfort level with that kind of financing and their comfort level with you as a borrower.
The more attractive you can make yourself as a borrower, the better. Most of that is going to come down to your Debt to Income Ratio. What's your tax return show you have made over the last two years? What does your W-2 say you make each month? Does that line up with what's being shown on your tax return? What out standing debts do you have? What are their minimum payments?
What the book may be referring to is the limits you'll eventually face using Agency Debt, Fannie and Freddie backed loans. A individual borrower may have up to 10 Fannie and Freddie loans under their name. Some married couples will even apply for loans individually so they can have 10 each.
Once you have used all those Fannie and Freddie slots, you'll either need to pay off a loan to open up a slot, or seek portfolio lenders (loans that are NOT backed by Fannie and Freddie) which typically have higher interest rates and can have shorter terms and amortization schedules.