Why is "Enrique", even though its 'r' position is at the middle of a word, pronounced as a double "r"? What are the orthographic rules you need to know to determine if an "r" must be pronounced as a double "r" or just an "r"?
The "strong R" (as in Rat) is spelled as just one r when in the middle of a word follows an L, M, N or S. As it's said in the comments, maybe M should not be considered because I can't think on any word with "mr".
Alrededor, Conrado, desratizar...
Another way to understand this is based on what sounds exist in Spanish. From Wikipedia:
The alveolar trill [r] and the alveolar tap [ɾ] are in phonemic contrast word-internally between vowels (as in carro 'car' vs caro 'expensive'), but are otherwise in complementary distribution. Only the trill can occur after /l/, after /n/, and after /s/ (e.g. alrededor, enriquecer, Israel), and word-initially (e.g. rey 'king'). After a plosive or fricative consonant, only the tap can occur (e.g. tres 'three', frío 'cold').
I didn't know why this happened, but there's an explanation on elcastellano.org which boils down to this: common names or surnames follow the same rules as every spanish word, but "single r" is pronounced like a "strong r" (like "ratón") when following "l", "n" or "s".
This is obvious when forming words from other that start with "strong r": "enrejar" (from "reja"), "enredar" (from "red"), etc. But also in names like Enrique or Israel, or adverbs like "alrededor".
Pronunciation of r and rr in Spanish:
The Spanish r is harder and more abrupt than the English r. It is similar to the rd sound in 'card'. Additionally, r is trilled at the beginning of a word and following l, n and s. The rr is always trilled.
Note that the hardness of the r sound often depends on region. I've seen some Spanish texts from parts of South America say the r sounds exactly like an English d--this is not true in many places.