Oh wow, thanks for the warm welcome, Dave!

I thought this was a place for people who are passionate about their laundry. And yes, I feel strongly about the ugly hideous crease that goes halfway down the back of the shirt and won't come out no matter what I try, and no amount of googling has helped me figure out the cause of this and what can be done to resolve it.
Hopefully others on this board will have some more helpful and friendly advice to share.

Recordaras, I agree that was a very rude response to your question. Most of the members of this message board are much more polite and welcoming to new members and if they don't have anything productive to add, they simply don't contribute. I, for one, would like to welcome you to the board!
In response to your legitimate question. Those creases are the reason I don't buy or wear permanent press shirts. They are permanent, due to the process of making the shirt "no iron". The shirts are in simplistic terms, coated with a polymer to keep them smooth. If there are any wrinkles in the process, they become permanent. On higher quality clothes this is generally done on the grei goods rather than the finished product. I'm surprised that Brooke's Brothers shirts have these issues.
Your best solution is to try to return them and buy all cotton and take them to the cleaners or iron them yourself. I iron. I too, hate that pleat pressed flat.
If the store won't let you return them, try customer service. For the price of these shirts, they should stand behind them. I might be naive, but I've had decent luck.

Aamassther, thank you so much for your detailed (and welcoming) reply!

In all honesty I had never seen anything quite like it, which is why I found it so puzzling. Back in the day, when my grandma taught me to iron men's shirts (and as an ambassador's wife she had plenty of experience in that field), one of her main rules was that there should never be any visible creases, under any circumstances. So I always attributed those to the drycleaner's sub-par method. And then I decided to surprise my fiance by hand washing and ironing all of his shirts while he is away to show him the huge difference that would make... only that didn't go quite as planned, and caused quite a bit of frustration.

This is definitely a lesson learned. I'm not sure we will be able to return his existing shirts, since he has had them for a while, but going forward we will never purchase "non-iron" again. I also noticed that there were several spots where the fabric started developing tears, only they didn't look like regular fabric tears with fraying at the edges, more like plastic cracking, if that makes any sense. Who would have thought?
I have already written a message to BB customer service, asking them to let me know which of their shirts do not have the permanent creases, and we'll start replacing the existing ones with normal cotton ones.

Once again, thank you for understanding and helping me figure this out. It's a shame that nothing can be done about the existing shirts, I was hoping there was some magic method that I just didn't know about, but at least I learned a valuable lesson.

Recordaras, I think talking CS is a good plan. It also makes them aware of your concerns and in my opinion, the defects in their product. I would also make them aware of those wear areas as well. That shows to me a cheap finish on an expensive shirt.
Kudos to you for trying to take care of your fiancés clothes. There was a time when we tried to make things last, and took pride in ours and our families appearance. Home keeping is an art and science that, from the sounds of it, you are willing to take seriously.

I am a male member and I like the pleats in the center back of my shirts. I wouldn't have another all-cotton shirt. Always need ironing and are a pain to iron so I send them to the laundry. That makes them even more expensive. I feel that no-iron shirts are a god-send. I usually buy J.C. Penny shirts because they fit and are a quality product. The back pleats give an ease in an area where my body needs it.

I also like non iron anything. I have not ironed in two years because I am only buying non iron clothes. Someone and I can't remember who had to have 3 creases down the back of his shirts, it was either military or police officers. They used to sell shirts with stitched seams down the back.

@recordaras - the BB all cotton button-down collar oxfordcloth shirts that are not "no-iron" may not have that crease set down the sleeve. My hubby has worn that shirt for years but we've always sent them to the laundry. He doesn't mind the creases on the sleeves and that's fine with me. They are a bear to iron at home, but if you are dedicated, it can be done. Ask me how I know ...

Those shirts hold up like iron and are very cool to wear in hot & humid summers here in The South. I worn them for sun protection out on the lawn tractor, when the cuffs are too far gone & worn out.** I was amazed at how comfortable they were, out there!

**I turn the collars to get extra miles on them and have even used the body of his older shirts to make new collars, but I draw the line at turning or replacing the cuffs. Once they're shot, the shirt becomes a donor shirt for the collar project.
We only buy the shirts on sale, and even at those prices, I maximize mileage from them.

Also, be aware that shirts bought at a BB "outlet" are different from the ones bought at a BB store. Been there, done that, and noticed the difference in fabric weave & shade of white.

@EmmaR - I don't have any experience with police uniforms, but the Army dress uniform shirt certainly has some... interesting creases. In all honesty we just trust that one to the dry cleaner, since it gets worn so rarely.

@Cavimum - thank you for sharing your experience! We've definitely noticed the BB vs BB "346" difference, and I'm sure it's even greater with regular cotton shirts.
I actually find the crease down the sleeve somehow less offensive than the crease down the back pleats - that one really drives me nutty for some strange reason. Probably because the sleeve one has a practical background - it's much faster to iron sleeves that way vs getting out the sleeve board, while the back pleat one just doesn't make any sense.

In any case, I already have a couple of orders placed for "non non-iron" dress shirts, and we'll probably keep a couple of the regular non-iron ones for travel.