Many people who want a horse can only keep one. Horses are herd animal, they thrive in the company of their own kind. As such, when keeping horses, it is important to provide them with a companion animal.
Goats and Sheep
Goats are a popular companion animal for horses in the racing industry. Many of these horses are kept in stalls for the majority of the time (particularly at tracks in Canada and the US, but not so much in the UK). Warmblood Horses also have a tendency to be kept stabled for most of the day. These horses become bored and can develop stable vices. In certain cases goats are provided to be a stall companion for a horse.
Even when horses are not kept stabled, but are on pasture, having a smaller companion is a good idea. Introductions may be needed by keeping the sheep or goat(s) in a pen where the horse can see them for a few days, otherwise the horse will likely be afraid of these smaller animals.
Fencing and predators are a concern in that goats and sheep can easily escape from most fencing used for horses, and predators can easily enter. As such it may be better to fence for the needs of the goats, or sheep.
In general if a horse is on pasture, even a pair of sheep, or goats, might work in terms of providing companionship. Both goats and sheep do best (even with a horse) with another of their own kind. Bottle raised animals will be the friendliest with people.
Alpacas and Llamas
(note the sheep fencing)
Alpacas, or llamas make better companions for horses on pasture, mostly because of their larger size. In this case, however, you may only want to keep one of either animal, either one alpaca, or one llama. Two is fine but they may become exclusive and leave the horse alone.
Llamas and alpacas are known for spitting, however a well handled, properly socialized animal will not spit, in fact, they can even be halter trained. Llamas and Alpacas do like to have a spot to roll but do not require a lot of feed, most do well on pasture in the summer, and hay in the winter.
Miniature Horses or Ponies
If a person cannot keep two full size horses, they may find that they have room for a miniature horse, or even a Pony. Miniature Horses, and Ponies, have quite a range in size, and full sized horses may be frightened of them at first, as will they must not have oats as they are very prone to foundering.
Donkeys and Mules
Although most people think of donkeys as being small, they can be big enough for riding. Donkeys and Mules are great companion animals, and bond to people very well. In fact it may be the donkey that approaches its owner even before the horse does.
Donkeys, and Mules to some extent, are also prone to founder and their ration should be watched.
Cats or Dogs
Can cats and dogs make companion animals for horses? Some breeds of dog may be prone to chasing a horse. The cats and dogs are free to come and go, and as such they are not really good for companionship purposes.
Note: Use special care when introducing a companion animal to a stud horse, many stud horses can be mean to smaller animals, even killing them.