What are the "Know Your Customer" challenges that Bitcoin faces? Can anyone explain what KYC (Know Your Customer) regulations are "missing", tough or impossible to implement with Bitcoin?

I will offer a bounty to the most detailed answer
(However this question is too new for me to do so)
TL;DR

Businesses in the Bitcoin space face no special challenges to implement KYC regulation, except the privacy oriented mindset special to some bitcoin users. However, it is impractical to enforce KYC regulation on the entirety of the bitcoin network itself.

Long version below

Know Your Customer (KYC) regulation applies to financial institutions and other regulated companies such as for example money transmitters and currency exchanges. It's stated goal is to make such companies report potential money laundering and to protect against identity theft.

The core component of KYC forms around following a Customer Identification Program, which means that the above introduced institutions have to form a reasonable belief that they know the true identity of each customer. This means that in order to comply with regulation they will require their customers to validate their accounts. The companies don't face any special difficulties whatsoever, except that there might be a slightly bigger number of Bitcoin users that will be adverse to their services due to KYC requirements.

On the other hand, companies that offer comparable money transmission services as the bitcoin network itself would be required to implement KYC regulation. It is possible that at some point in the future a country might require by law Bitcoin users to disclose their bitcoin addresses. However, that would be impractical – if not impossible – to enforce, as, due to the decentralized nature of the Bitcoin network there is no central entity that could be forced to disclose such information about other users and there would be any number of participants that would have to be monitored and brought into compliance in order to collect such address information about their business partners. Additionally, even if associated to certain addresses through business partners informing on them, in most cases users would be able to plausible deny control over said addresses.