Well, you could do without agents but you'd really be handicapping your game. Here's how I'd rate them, from least to most important:

Princesses are probably the least useful, as evidenced by the fact that the Muslim factions get by without them. They're just another diplomat with a bonus of getting a free general/family member when they marry.

You could do without Merchants. They're a nice money-maker, and can be used for economic warfare without repercussions, but they're not essential, especially if you're playing a faction in a part of the map that has good income from trade anyway.

Assasins are useful, but not essential. I use them heavily, but it's more for the fun factor, and because it speeds up conquest when I cause economic damage with sabotage, and take out enemy generals before a battle. I could do without them if I had to.

You could do without diplomats, since other factions will come to you with proposals. You'd lose out on early income when getting started by selling map info, and you wouldn't be able to bribe away armies later in the game.

Priests/Imams are very important. You won't have any influence at all over the College of Cardinals as a Catholic faction without them, and you'll need them to control heretics and witches. They're also important for converting local populace when expanding into Muslim lands (or vice-versa, playing a Muslim faction). There are cities on the frontier of my Spanish empire in Africa that I couldn't hold onto, without priests in the advance guard to stabilize the population and prevent religious unrest.

Spies are the most important agents (IMO), not just for strategic intelligence like finding out what units are in a city you're about to siege, but also because other factions will use spies and assassins against you. They're the only way to root out enemy spies causing unrest in your cities. They can also be used to cause unrest in enemy cities, or even "friendly" fellow Catholic cities you'd like to flip to rebel status, so you can take the city without angering the Pope. Spies are very powerful... I crank out as many as I can afford, paired up with assassin companions.

I could play the game with only priests/imams and spies, and no other agents, but I think I'd have to use a lower campaign difficulty setting to balance out the loss. Remember that the campaign AI is going to use all these agent types against your faction (one way or another), and in many cases the only way to counter an enemy agent is with an agent of your own.