This is a very interesting question, and I'm enjoying reading everybody's responses. 
As for me, I burned them 15 years ago in a BK7, and I was fairly sure they woudn't approve me, but I wanted a free copy of my credit report/score so I applied for the Green card... and almost fell off the chair when it was approved! 
I had already been reading the boards here for months, so I was aware that many times a charge product and a credit product can be applied for in the same day with only one HP, so I figured the BCP was the card that would really fit my spending, so I app'd for it, and was approved as well. I nearly needed my inhaler.
Next year when the AF is due on the Green card, I plan to cancel it since (honestly) I really didn't want it to begin with. I don't have a good use for MR points, and my credit lines are high enough on my no-AF revolving accounts to allow me the flexibility I need. As for the BCP, even after you subtract the $75 AF it is still making me far more in cash rewards for my spending pattern than any other card out there (at least, ones easily available to me).
Hope this helps!
I have the Delta card because I travel on Delta and its international mile share partners. I get offers from AMEX for their other products quite frequently but don't want any. AMEX is not a lender I want a relationship with, unless I have to. I hope Delta dumps them soon just like Costco did. 
@Anonymous wrote:
Why did you choose American Express? How does it benefit you? Which type of card do you have? Do you use the benefits enough so that the yearly fee is worth it?
I don't know why I chose it. I'm not thrilled.
Have an Amex business platinum which charges $450 per year, extends no credit, does not report to credit bureaus, and has perks I never use.
And have 2 Amex credit cards, which have no balance transfer offers, haven't given me a CLI despite 13 months of perfect handling of all accounts, and which both have annual fees.
