Can I have a pet monkey?

As much as you can have a pet alligator or a pet rhinoceros.

Forget it.  Monkeys are classified as "exotic animals" which would ban it.  Service animals are specially trained to perform certain tasks that a person cannot do itself.  Your landlord would not consider a monkey a service animal.  Monkeys are not for the average person.  They require special housing (cages), food, and have to wear a diaper because they cannot be potty trained.  Adult mature monkeys will be more likely to bite you which could be dangerous.  Very few vets will provide vet care for your monkey.  

Just forget getting any kind of monkey.

I guess you can, just remember it's no joke.  You might need to find a vet who specializes in the care of wild animals.  You might also need to live somewhere where the monkey will have plenty of trees to climb and space to run and play.  And then you might need to pay for services to properly housebreak the animal and care for its grooming, hygiene, dental care, etc.  You might try researching about the restrictions in your area and reaching out to others who have wild animals as pets.  Consider though that it might be unlawful to keep a non-domestic pet under the pretext of keeping it as a service animal if no one in your family is actually disabled.  And then if some innocent person/child gets seriously injured or maimed by the monkey you might have a major liability on your hands.  I would plan keeping such a pet and saving enough money for it at least 6 months in advance until you are absolutely sure you know what you are doing.

The ONLY animals that can be service animals are dogs and mini horses.

No, you cant have a monkey as a pet.   If you live in city limits, you likely do not have a proper outdoor enclosure for a monkey.   

Forget about what everyone else says, do what you want.

If you wanna have a threesome with your spouse and a chimp, be yourself.

No, there are not magical loopholes that would allow you to have a monkey somewhere that prohibits their keeping. Service animals are restricted by federal law in the USA to task-trained dogs for the disabled and miniature guide horses for the visually impaired. 

Additionally, it is very cruel to keep primates as pets. Their complex social needs cannot be adequately met by humans. They belong in the wild with groups of others their own kind. 

Monkeys, ALL monkeys, are wild animals, and it is cruel and inhumane to keep one as a pet. Get a nice dog, thousands are euthanized every day. Save a life. Let the monkeys live in the wild, don't support their abuse.

Monkeys are a huge commitment. More than children.

Don't.  They are dirty and will pee on you when you're sleeping. 

Rescue a dog. Forget about monkeys.