Hjörtur Gísli Sigurðsson sources penises from farmers, slaughterhouses and trophy hunters. Occasionally Iceland's Marine Research Institute will call him to say that a whale has washed up on their shores. Sigurðsson and his associates rush to the spot to harvest the creature's organ. The collection of the Icelandic Phallological Museum has swollen in size from 63 specimens when it opened on August 23rd 1997 – the birthday of Sigurður Hjartarson, its founder and Sigurðsson's father – to 286 today.

This year the museum is on track to draw 50,000 visitors, up from 14,000 in 2011, when Sigurðsson took over as curator from his father. It remains a family enterprise: Sigurðsson's wife and son work there full-time and his son-in-law comes in part-time. Visitors pay 1,500 krona ($13) to get in. About three-quarters of the revenue comes from ticket sales and the rest from the gift shop, which does brisk business selling, among other things, condoms and penis-shaped salt and pepper shakers. Sigurðsson says it has been profitable for the past two or three years, boosted by word-of-mouth and the dramatic rise in tourism to Iceland. Foreign tourist arrivals to the isolated north Atlantic nation have doubled since 2012, to 1.3m last year. It helps too that the museum is in central Reykjavik – and is the only one of its kind in the world.

Unlike the Sex Museum in Amsterdam, a tawdry tourist trap, the phallological museum is neither lewd nor pointless. Sigurðsson says schools organise trips to the museum, and that parents often bring their children along. Children under 13 are not just allowed in but can enter free of charge. It is easy to see why: the experience is, perhaps unsurprisingly, educational. The collection includes specimens from 22 types of Icelandic whale, seven varieties of Icelandic seal, 20 Icelandic land mammals ranging from the house mouse to goats, and 34 foreign mammals, including horses and elephants. In a delightfully Icelandic twist, there is also a small "folklore" section, featuring the male reproductive organs of, among others, the corpse-eating cat of Thingmuli, the nasty ghost of Snaefiall, and an Icelandic elf, or "hidden man". The last of these display cases is, ingeniously, empty.

Iceland has a fertile culture of mythology, and chief among its myths is that of the elves – creatures that look like us, live in rocks and are only visible to true believers. Sigurðsson says that some visitors to the museum do indeed see the "hidden" specimen in the cylindrical display case. It is mostly women who see it. Sigurðsson offers no explanation beyond speculating that perhaps women are blessed with more active imaginations than men.

The majority of the collection lives less in the realm of fantasy (except perhaps in the weirder corners of the internet). It is fascinating in its diversity – the male member varies hugely across species, and some specimens are frankly baffling. How precisely do the mechanics of a duck's corkscrew work? Why do some animals have a large penis bone – called the baculum – and others a tiny one? Why in some species, notably humans, is it missing altogether? And how on earth did such a museum get started anyway?

The final question is the easiest for your correspondent, a journalist and not a biologist, to answer. In 1974 Sigurðsson's father, who was at the time the headmaster of a secondary school, received a bull's penis to use as a whip. (The founder goes into no more detail, on the website, about why he was given this present.) At the time he lived in Akranes, a small village near a whaling station where some of his friends and colleagues worked. They started to bring him whales' penises, supposedly as a joke. Over time he started to collect them. "He has a very hard time throwing things out," says his son. In 1997 he decided to open a small museum.

The legend of the place has grown since. In 2012 a pair of Canadian film-makers released a documentary called "The Final Member", about Sigurðsson's quest to add a human penis to his collection. The previous year the museum had at last acquired one, from an Icelandic donor named Pall Arason, who described himself in his letter of donation as a "worthy disciple of Don Juan the noble Spaniard and other renowned genii of history". This specimen is now on display. At least four more men, including a German, an Englishman and an American, have pledged to donate their organs to the museum. (However the museum still doesn't possess a penis from the narwhal whale or the Atlantic white-sided dolphin.)

Sigurðsson is thinking about expanding the place, making use of the building's basement. If he does, the museum would double in size. It is growing in ambition and influence too. Sigurðsson says he is sending a horse penis to a show in Norway in December, and that he sometimes trades penis bones with collectors in America and Europe. It is five years this month since he took over the museum. If there is one thing that remains constantly surprising, he says, it is the sheer diversity of specimens even within the same species. Each penis, like the museum itself, is unique.