A former Professor at JTS used to daven in a very Charedi shul in the Shaarei Chesed neighborhood of Yerushalayim. He explained to surprised people that "the people I talk to I can't daven with [because they don't daven according to halacha] and the people I daven with I can't talk to [because of the gap in hashkafa]. I can live without talking. I can't live without davening."
Oh that we all felt that we can't live without davening! Rav Soloveitchik was bothered by the question of how it is PERMITTED to daven. How can we trouble Hashem with our petty requests. Like when I was a kid and I would BEG him to help my Mets or Dolphins win. OK - that was pikuach nefesh... But how can we daven to Hashem for a few more steaks or to be able to lease a new car [i.e. money] or other things that occupy our often small world?? [That prompted him to explain that תפילה needs a "מתיר".]
He was also bothered by the fact that when we stand before G-d we are liable to be swallowed by His Presence! [This doesn't bother us because we don't feel His Presence so strongly. But if we did...] So how do we have "the guts" to daven?!
Rav Soloveitchik's answer is that we daven because we CAN'T NOT DAVEN!!!
WOW!!