I did study a bit of complexity theory back in the day, and it does seem a bit theoretical, but not so long ago, I ran into an very prominent manifestation. To help someone, I had created a simple web page which also had an input field for the css that was applied directly to the page. To avoid flickering and weird effects because of malformed css, I wrote a regexp to check the CSS code. It's a very simple language, after all, and it didn't have to be perfect. It worked well, until at one point, a simple typo locked up Chrome for 1 minute 10 seconds. Parse time just jumped from less than 100ms to 70000ms by adding a single character. That's when you feel what exponential means.

I just want to point out that complexity theory isn't really about analysing the complexity of a particular algorithm (that's the subject of algorithm and data structure analysis, where students learn big-O notation), but about understanding the complexity of a problem independently of any algorithm (i.e. what problems couldn't be tractably solved by any algorithm).