My process... I sit down at the piano and play. Frequently. I never play the same thing twice during these sessions. All over the map. Clears my mind. Then, I'll walk away and do something else for a while, often for days.
Sometimes, though, later that night (usually on a Friday), I say "hey, I'd like to write a song about (blah)." This is generally midnight or later. By the time I start writing, I'm about half asleep, and this appears to be the key....
The next phase involves me sitting down at the keys and just suddenly brain-dumping a melody, chords, and words, frantically scratching down the words on a piece of paper every few lines. This process continues for up to a half hour. Then, I fall asleep.
The next phase occurs in my sleep. The song tends to haunt me and the next morning, I awake feelling like I've been up all night writing, even though I haven't....
The next phase occurs at some point later in the day, usually in the morning. This is when I play through the song from the night before, sometimes playing different chords for parts because I don't quite remember them from the night before. I often realize that the melodic line and/or chords for some part of it suck, go back, and rewrite bits of it until I'm actually moderately happy with it.
The final phase is the performance phase. I perform it repeatedly---at least once a day. This is the most important phase, as the melodic line evolves slightly over time, and occasionally, I change a chord here and there (an 11th here, a 13th there...).
Then I record.
Maybe it's just me.