Why is Antirrhinum written with two 'r'?

In Greek compounds, if a stem that begins with rho is preceded by an element that ends in a simple vowel (not a diphthong), the rho is doubled. Likewise, in inflected forms where a simple vowel is added before initial rho. Or, as Smyth, Greek grammar §80 puts it:

An initial ρ is doubled when a simple vowel is placed before it in inflection or composition. Thus, after the syllabic augment...,  ἔ-ρρει, was flowing from ῥέω; and in καλί-ρροος, fair flowing. After a diphthong ρ is not doubled: εὔ-ροος, fair flowing.

In §80.a, Smyth explains that this doubling of the rho is 'due to assimilation of σρ (ἔ-ρρει, καλί-ρροος), or ϝρ (ἐρρήθη was spoken).' Although this doubling through assimilation is 'strictly retained in the interior of a word,' it's 'simplified to single ρ when standing at the beginning, i.e. ῥέω is for ρρέω.' When the rho in a compound is preceded by a diphthong, as in εὔροος, the doubling doesn't occur 'due to the influence of the simplified initial sound.'

As for accent, in Greek, the accent would follow the normal rules for accentuation. For compounds, the general rule is given in Smyth §178:

In composition the accent is usually recessive...in the case of substantives and adjectives, regularly in the case of verbs: βάσις ἀνάβασις, θεός ἄθεος, λῦε ἀπόλῡε.

Therefore, in Greek, the accent of the noun ἀντίρρῑνον, 'calf's snout,' is on the τί syllable, and the iota in the stem is long.

However, in modern languages, the accentuation of words doesn't always follow the rules for the ancient languages. (There's also the fact that accent in the ancient language and in modern languages is of a different type: pitch vs. stress). In this instance, Merriam-Webster tells me that the modern word antirrhinum is pronounced ˌan-tə-ˈrī-nəm, with the primary stress on the (r)rhi syllable, in which the i is long.

For the doubled "r," that is a spelling convention dating back to Ancient Greek, inherited by Botanical Latin - preferred, but optional. Reference Botanical Latin (4th ed.) by William T. Stearn (p. 261).

For Botanical Latin, any consistent, intelligible style of pronunciation is acceptable, with some preferred conventions. Refer to Botanical Latin (pp. 49-54). I suggest antiRRIHnum long "I" in the stressed penultimate syllable.

Incidentally, I highly recommend the above cited book as a best friend for understanding any aspect of Botanical Latin. It is a thorough, well researched, and very charming book on Latin in the world of botany.