Currently: (1) not having adopted the full legal framework, (2) failing to win the support of the populace of members that have referendum on the issue, (3) being caught up in the EU's institutional bottleneck (e.g. the EU structures are currently set up for a maximum of 27 members, which we reach next year, only a new framework like the one proposed in the Constitution can allow more). The rules may change. Conservatives certainly want so, adding cultural, religious or geographical criteria. There is a broader opinion to add time and size limitations based on the EU's ability to cope with assimilating new members.

if they whole world decided it wanted to join the EU, on what grounds would you deny them admission?

See above. Currently, the EU could theoretically grow to encompass the whole world.

Or would you prefer everyone were allowed in?

Personally, I would very much endorse an explicitely open-ended EU, albeit one growing slowly (say, integrating the current 27 in the next ten years, absorbing ex-Yugoslavia and Turkey over the following ten, Ukraine and Morocco over another ten, Russia, Syria, Lebanon, a peace-agreed Israel/Palestine over the twenty years thereafter...)

But another possibility I would like is one Colman hinted at, that EU-like organisations start to grow elsewhere. I note there are some already: ASEAN and the African Union were in part modelled on the EU, they look like earlier stages of European integration, and could develop into something similar to what we have now over decades. If such organs of regional integration develop, it would be best if instead of confrontation, there could be some overlap, say Morocco being a member of both the EU and the African Union. This would be nothing radically new: think for example of Norway, which is part of a a closer Scandinavian cooperation and customs union, but not an EU member.

And if so, would you continue to call it "The European Union?"

I don't think names matter that much. It could be changed to "Euro-Mediterranean Union", "Eurasian Union", "Yellow Stars On Blue Union"... Such a name change wouldn't be the first: after all, the EU is called the EU only for one-and-a-half decades, before there were names like European Community, the Common Market, Montanunion...

_{*Lunatic*, n.
One whose delusions are out of fashion.}