Albert Einstein's general relativity was developed from 1907 to 1915, not in the blink of an eye. Yes, it is unlikely that any of us will ever encounter any problem which would require thinking of that deep and sustained nature, but that does not mean we will not combat and conquer only easily solved problems.
When I need to do my heavy thinking, I go to sleep. More precisely, I lay down in bed and let my mind take a meandering path through the problem space. I uncouple the 'obvious' linkages within the problem and see what happens when I couple items which would normally make an unimaginable combination. Its during these times of free association that I regularly hit upon a solution to my most difficult issues.
When I read a blog post by Paul Graham about The Top Idea in Your Mind, I instantly understood where he was going in reference to his morning shower. His best thinking is done there, just as mine is done in the minutes (and sometimes hours) between laying down and falling asleep.
As a child, I was raised in a Baptist church. One of my youth ministers claimed that reading the Bible as the first act of the day was vitally important for everyone and is something he wanted all the youth to do every day. As a means of 'encouragement', he once set up a phone tree so that the youth such as myself who were NOT morning people would be called every 2 minutes by the youth who were morning people to remind us to get out of bed.
I bring up that story as a way to show that sometimes even the best of intentions cannot produce quality results. Just like I could never do quality thinking in the morning like Paul Graham does, it is unlikely that he could ever do his deep thinking as he fell asleep in bed.
To give a fair assessment to problems my stakeholders bring, I generally ask them to let me sleep on it first so I can give them the best answer possible. My stakeholders know this about me and I believe they respect me for it, mostly because they keep coming to me for answers.
What about you? Where and when do you do your best thinking?