This is how I understand the situation (I'm not a lawyer, nor a DAR!). This is just a result of my research on this problem.
There is no exception to the rule that requires a registrant to be at least a permanent resident of the US (i. e. a green card holder). However, there are two workarounds here.
First, according to FAR 47.9, a corporation which is not a US citizen (i.e. established by a non-citizen), can register an aircraft in the US, provided that this aircraft is operated inside the US for at least 60% of the time. Now, I'm not sure that someone who is not a tax resident can establish such a corporation at all. I'm not saying it is impossible to do, I just didn't look into it. Then, I've talked to a lawyer who said this was not legal to establish such a corporation with a sole purpose of owning an aircraft. However he gave me no references at to why this would be illegal. And I've talked to another lawyer, who said it was perfectly legal. So I don't know. I've heard of people going this road, although not with experimental aircraft. The upside of this seems to be that you (through your corporation) maintain the full control over aircraft registration, compared to the other approach.
According to FAR 47.7, in addition to citizens and resident aliens, another entity that can register an aircraft is a trustee. This is used mostly by people who live abroad, but wish to have an N-registered aircraft (that's because US general aviation laws are much more relaxed than European, say, and one can fly a N-numbered aircraft with a US pilot's license). In this case, the buyer of the aircraft trusts the aircraft (via a trust agreement) to a trustee company. The buyer is also listed as an exclusive beneficiary in this trust agreement. So, from the point of view of the FAA, the aircraft is registered to a US trustee, which is basically the biggest downside of this approach for you. Even though the registrant of the aircraft cannot seize it from you (because there are provisions in the trust agreement that disallow that), if something unexpected happens to the trustee company, I imagine it will be quite hard to re-register the aircraft to someone else (normally you need to acquire a bill of sale from a trustee company). There are provisions for this in trust agreements, but, again, enforcing this will mean some unpleasant time, I'm sure. I'm not aware of any incidents of this nature so far. Anyway, this is a much more popular route, and there are hundreds of aircraft registered this way. This is also officially recognized by FAA and there was a clarification on this topic from FAA earlier this year (this schema is mostly used for corporate aircraft I believe). There are also companies that specialize in this and will do most of the work for you for a reasonable fee. I personally am going to go this route and hope that it goes smoothly until I obtain my green card an can re-register the aircraft in my name.
I'm also not sure how this all works when you initially register an aircraft and obtain an AC as a builder.
As to finding someone to register an aircraft for you, I wouldn't do it. It seems that it more or less gives the control of the aircraft to this person. In addition, this person seems to be exposed to legal actions as an owner, should something unpleasant happen. I think I also read somewhere that this was illegal in the first place, but I don't remember enough details.
