Other than horses, elephants have been historically ridden into battle.

Elephants in antiquity were used to great effect on the battlefield. A number of ancient Greek and Macedonian generals used elephants on the battlefield, but perhaps the most famous general to use elephants was Pyrrhus of Epirus. The advantage of elephants was described by the historian Zonarias, who in his account of the Battle of Heraclea (280 BC), states:

… Pyrrhus raised the signal for the elephants. Then, indeed, at the sight of the animals, which was out of all common experience, at their frightful trumpeting… both the Romans themselves were panic-stricken and their horses became frenzied and bolted, either shaking off their riders or bearing them away.

Pyrrhus also used elephants at the battle of Asculum (279 BC), the battle from which we get the term "Pyrrhic victory".

Another notable use of elephants comes, of course, from Hannibal Barca. Taking notes from Pyrrhus, Hannibal brought 40 elephants along with him in his campaign against the Roman Empire. At the battle of the Trebia (218 BC), the historian Livy notes that:

Ad hoc elephanti eminentes ab extremis cornibus, equis maxime non visu modo sed odore insolito territis, fugam late faciebant.

Moreover the elephants, standing at the outer edges of the wings, terrified the horses greatly not only by unfamiliar sight but also smell, and brought about widespread flight.

However, war elephants did have their pitfalls. Livy recalls that Roman troops eventually found out that they could divert and fend off the elephants by firing arrows into their hindquarters, sending them charging through Hannibal's own troops.

SOURCES

Jeff Champion, Pyrrhus of Epirus (2009)

Livy, Ab Urbe Condita, Book XXI

Though not quite as big or fast. as a horse, camels don't need to be fed and watered as often and can handle harsh conditions much better. I'm not sure how long they've historically been used in battle (I'm sure their use goes back to at least the Middle Ages), but an example of recent war camels are on this page.

As a follow up, I know some horses are about twice the weight, and thus greater strength, compared horses. There must be some balance between size, speed, and amount of care and feed required which prohibited larger horses from war. How large were actual war horses? I'm really wondering if the horses deployed were smaller than camels, as I've never heard of a horse mounted gun.

Edit: It would seem I've forgotten about bactrians, and spelling.

I seem to recall reading something at one point about Canadian cavalry (mooselry?) trying to ride moose, but I can't remember much about the outcome and certainly don't have any sources. Does anyone know anything about that?