If you go looking for longish snippets of text that only use simple sentences, you'll find it surprisingly hard.
The fact is that simple sentences are quite limiting on their own, and putting a lot of them together can make a paragraph sound jarring or repetitive.
But this can be useful sometimes. Here's a snippet from The Martian, where the main character is mentally checking through some equipment to figure out what he can use.
(The section in brackets is its own sentence. The brackets are used instead of a full stop to show it's a related thought.)