The only fish that I recall giving me a feeling like that was a huge trout I hooked off of the dock at Lazy Valley on Taneycomo years ago. We aren't sure how big it was (probably not as big as my current PB 27" 10# rainbow trout that came from Taneycomo), but it was definitely a big fish. It broke the surface once and looked gigantic! After peeling a bunch of line off, it finally started heading towards me. I thought we were finally going to land this thing! Well due to my lack of skill at fighting fish at the time (I think I was around 10), the fish made a dash under the dock and got me wrapped up on something. If that happened nowadays, I'd just shove the rod down in the water as far as I could. The fishing method and gear was far from glorious as I was fishing an air injected night crawler on a Zebco rod with the double trigger guard handle and a Zebco 33 push button reel with 6# Stren Magnathin, but I still remember that fight like it just happened yesterday instead of around 20 years ago.
I did lose a 4# Gasconade largemouth in my trolling motor last year during a tournament. Aside from that, I really don't recall losing many big fish at all during my fishing career.