And the biggest influence on sound quality is...

1. Attitude. I've been moved to tears listening to music on a cheap transistor radio.

2. Speakers. Assuming you have a reasonable set up (whether inexpensive or otherwise), better speakers will make the most obvious improvements.

3. Room. I'll venture to say most people have little control here, and are unwilling to move their speakers or spend enormous sums on room corrections. For those who are willing, a good room can make a mediocre system sound good and a good system sound great.

Brain!  Can't say how many times I've just been driving down the road listening to the mediocre system in my pickup truck enjoying the heck out of music.  Ear/brain adjusts to quality. 

That's not to say audio quality isn't enjoyable.  But I would hope most folks wouldn't turn off a mediocre radio if the tunes were good! 

Disregarding the quality of the recording and considering some people have no choice on the room, strictly speaking equipment wise, I have to go with the amplifier.  I have found just playing around and substituting different components in a system that a very high quality amplifier will make even some of the worse speakers sound so much better at least to their dynamic and frequency limits.

Over the years and upgrades, I have found the preamp upgrades I have made to have the most significant effect on SQ.

Approximately 50% of the sound that reaches your ears in a residential sized room is from room reflections. I've heard systems comprised of nothing but superb quality, well matched components sound horrible in bad rooms. And systems composed of mediocre components sound engagingly musical in good ones. All depending on the room and setup (positioning of speakers & listeners). Naturally some recordings are disasters that sound bad regardless of anything else, but in a good room can still deliver a taste of the artists.

If your goal is to hear performances where you get a real sense of the performers & instruments, deal with the room first.