What is the word for a fear of hippopotamuses?

You better fear the hippos! Hippos are considered the world's deadliest large land mammals. They are very aggressive and territorial. They are also the most dangerous animals in Africa (after mosquitoes).

There is the term hippophobia for the fear of horses (and other hoofed animals like ponies, donkeys, or mules) as you mentioned. Hippo- is from Ancient Greek ἵππος (híppos) - horse or relating to horses. Equinophobia is a synonym (from Latin equīnus - of or pertaining to horses).

Hippopotamus literally means "river horse":

From Latin and New Latin hippopotamus, from Ancient Greek ἱπποπόταμος (hippopótamos), from ἵππος (híppos, “horse”) (English hippo-) + ποταμός (potamós, “river”). - Wiktionary

There is no established term for the fear of hippopotamuses but it would be a rare case of zoophobia. Hippopotamophobia is used as a neologism though and appears in some online sources.

From Wikipedia's zoophobia article:

Zoophobia is not the sensible fear of dangerous or threatening animals, such as wild bears or venomous snakes. It is a phobia of animals that causes distress or dysfunction in the individual's everyday life.

There is also a more specific phobia called agrizoophobia which is the fear of wild animals. Most of the sources say that it is rarely diagnosed and it usually effects people in urban lifestyle. Zoos and films are common triggers.

From the book "Anxiety Disorders in Adults A Clinical Guide" By Vladan Starcevic, MD, PhD:

While the reason for fearing certain animals lies in their dangerousness (e.g., sharks, lions, crocodiles), many dangerous animals are not objects of an animal phobia. This is because most people afraid of these animals are not impaired by such fear (e.g., they simply avoid places where they might encounter dangerous animals). Indeed, patients with animal phobia are usually afraid of animals such as snakes, dogs, cats, spiders and other insects, rats, and mice. Only a minority of these animals is dangerous, so factors other than danger seem to be more important in determining whether particular animals will be feared and avoided to the extent characteristic of a phobia. The most important of these factors is a feeling of disgust elicited by some animals (e.g,, Tolin et al., 1997; Lipsitz et al., 2002). It has been argued that with animals such as spiders, the main underlying issue is a feeling of disgust rather than a perception of danger, considering, that only 0.1% of all the varieties of spiders are dangerous to humans (McNally, 2002).

The word hippophobia refers to a fear of horses

This is not really so. The first "-phobia" was hydrophobia (now replaced by "rabies") It refers to an irrational and overwhelming fear of water categorised as a mental condition as a result of a medical condition.

1547 A. Borde Breuiary of Helthe i. f. Cxliv Hidroforbia..is abhorringe of water... This impediment doth come..of a melancoly humour.

The OED notes

this [hydrophobia] is probably the model for subsequent English formations. Formations within English are found from the 17th cent. (in an isolated example); a handful date from the 18th cent.; from the 19th cent. *they are very abundant. Combined with a wide variety of first elements: these may be ultimately of Greek or Latin or English origin.

We find the Victorians inventing innumerable "phobias" and (with the popularity of psychology) a continuance of this into the 20th century and present but now in a satirical or humorous way, and thus a gradual weakening of the "fear" aspect:

From the OED

Forming nouns with the sense ‘fear of ——’, ‘aversion to ——’.

1803 A. Seward Lett. (1811) VI. 94 He is a very laconic personage, and has upon him the penphobia.

1824 Life (1849) I. 125 She laboured under a perpetual dustophobia; and a comical disease it was.

1843 Blackwood's Edinb. Mag. 54 245 That powerful..writer..depicts the same regiphobia as raging among the Parisian Charlatanerie.

1896 Westm. Gaz. 6 June 2/2 The cycling craze has produced the antagonistic disease of cyclophobia.

1902 Westm. Gaz. 21 Oct. 2/3 There were symptoms in the City attitude of a certain amount of L.C.C.-phobia [= dread of the London County Council].

1928 A. Huxley Let. 9 Nov. (1969) 304 When my epistolophobia becomes..acute I will apply to you for..aid.

1978 N.Y. Mag. 3 Apr. 85/2 (advt.) Swim-o-phobia? Cure it forever. Our private lessons by professional instructors will have you phobia-free and swimming in no time.

1997 Independent 25 Mar. 119/4 It was immediately obvious that screening isn't very accurate, that an abnormal smear is rarely anything to worry about but we've sown the seeds of Big C-phobia anyway.

Four things emerge from the above:

1. Not everything has a phobia attached to it.

2. The original seriousness of the formation has been lost.

3. A fear of hippos is entire natural and unlikely to be capable of being a phobia.

4. Under the rules of "fair use", you can use any reasonably descriptive or defining prefix to phobia to create a new "phobia".

If you seriously think that there is a need for a word for "a fear of hippopotamuses" (above and beyond any natural fear) and that it is a genuine category (I am rolling my eyes here), then following the pattern above, there would be nothing wrong with "hippopotamusophobia"