EURid (crazy name, crazy guys...) has been kind enough to publish stats for registrations to date of the .eu top level domain by country / territory, and Germany emerges a clear leader based on raw volume - 1,041,614 so far. Bottom of the heap is French Guiana at 59, although it is possible that there are other special cases where no registrations have been made - St Pierre & Miquelon for example. We've managed just shy of a third of a million.
Anyway, so far so not very interesting. I have had a go at charting .eu registrations relative to population, and it is Gibraltar which emerges as by far the most enthusiastic, with 3,370 registrations for a population of 31,000. This, I would imagine is down to UK entities with a Gib presence preferring .eu to .gi, thinking it less likely to scare the horses. Of the EU 27, Cyprus leads from Luxembourg and the Netherlands.
At the other end of the scale, of the sovereign states it is the Spanish, Portuguese and the Romanians who have the lowest level of registrations per capita. However, it is France's Dom-Toms - the recipients of quite phenomenal levels of EU largesse - that are the least enthused. Still more hopeless stattos can find figures for net donors and recipients per capita here.