Dexus, a company that manages real estate only in Australia has $26 billion of property under management, split about 50% owned by them and 50% managing for other people. Their total portfolio is under 150 properties. That's all in a country with 20 million ish people. You could get a lot bigger than that.

But of course it wouldn't be the quickest way to grow since you'd need insane captial. If your competitive advantage is the extra return you're squeezing from the properties, makes far more sense to lease them so you can have more, rather than be buying them upfront.

https://www.dexus.com/who-we-are