Yes, Hopsital would have allowed us to keep the placenta
I delivered in a community hospital, enroute to the city hospital we had planned to deliver our twins at. We could have taken the placenta home.
I had been working/arguing over a mile-long birth plan with my city doctor for some time, so they knew that I might want to take the placenta home. My city doctor faxed my birth plan to the community hospital when it was suspected we would be unable to make the whole drive, but the attending doctor had not had time to properly review it prior to the delivery.
After the delivery, she further read our birth plan, and came to me to clarify "I may want to take my placenta home". (I had never firmly decided, but also thought that this inclusion might discourage them from early cord clamping - which we prohibited, but was done anyway - if the hospital routinely collected cord blood).
I decided not to bother taking the placenta home. I had wanted to perhaps make some prints with it. But buying several sheets of large, art-quality paper was one of those little things that just didn't get done prior to the birth. I wasn't interested in freezing it until spring so that I could bury it. So I said they could just get rid of it.
Now I'm really sorry that I did that. We did look at the placenta after the twins were born. But I miss not having an opportunity to really scrutinize it after some rest. And I'm sorry I didn't figure out the complicated logistics of quickly getting some art paper from the city after the birth and making those prints.
I would imagine that this is one non-mainstream request that is pretty easy to accomplish with most hospitals. I doubt anyone at the community hospital we birthed in had EVER asked to do this, but they treated it very normally. (On the other hand, they might have been so astonished by the litany of "crazy" ideas in my birth plan that this seemed pretty innocuous!

As far as that goes, "crazy" was basically "let it all go naturally even though it's twins and Twin B is breech")