Not a fungus. Instead a kind of cottony scale insect. The white stuff is an egg case which you need to destroy.

Would it be too drastic to cut the stem where there aren't any branches coming out of and most of the scale are growing? Otherwise, how can I destroy the scale egg cases effectively?

If you mean, cut the plant all the way back to the ground....that is a method of control that is often suggested.

You'll need to find control measures for your plant(s), though. The insects will return to infest the new growth.

If it's that's infested, yes cut it back. Unfortunately I know of no fast scale predators. There are excellent controls, but they can take some time to work, and with a heavy infestation you may not have that time. A large enough release to try and make up for the size of the infestation would be expensive, and might not be successful if you don't get the right predator for the type of scale you have.

What I would do:
Cut back tree
Get a definite ID on the type of scale you have - this may require some very close-up macro shots on what is a pretty tiny insect. If you don't have a camera that can get a small enough depth of field, see if you can find an entomology school near you, or check Bug Guide for people who may take a specimen for ID.
Once you know what you're dealing with, buy the best predator for it and make multiple releases in your yard over the growing season. It may take several years to build up enough of a stable enough population that you could relax your guard a little.

You may be able to do without the definite ID, but because its scale and the predators for scale can be specifically selective sometimes, I would get one.
#1 predator - Ladybeetles! Some good species to try (commonly used in horticulture around the US):
Cryptolaemus montrouzieri (primarily a mealybug predator, but they like scale too). aka Crypts
Rhyzobius lophanthae, aka Singular Black Ladybeetle
Cybocephalus nipponicus - not really a Ladybeetle, but in the same Superfamily
And a parasitic micro-wasp: Metaphycus helvolus

All of these things will take time to take hold and take a population down, as I said. They are preventatives and maintenance once the main infestation is gone and you need to keep up with them once scale starts in your yard. It's a tough pest to deal with.

Oh, whatever you do use, don't use common Asian Spotted Ladybeetles - Harmonia axyridis. They fly away from your yard after you release them, so do little good in the end. They're great to have in the environment in general, so are good to release as a general environment beneficial, and eventually you may see them and their larvae come visit a problem area you have, but that's it.

bake, you are probably over doing it for what 99% of the general population will likely do to control their pest insects. This scale is clearly one of the 'cottony scales'...and it probably doesn't matter to valvedsteel the EXACT genus and species.

A very effective method of control for scale insects has historically been and still remains today....horticultural oil. Our original poster could probably gain control over his pest problem with proper applications of oil without cutting his plant back to the ground, though I would be sorely tempted to begin some prophylactic treatments on brand new growth.

Encouraging a multidimensional environment is always extremely helpful in inviting in the 'hoards' of beneficials to help US keep the pests down.

Here are some more shots that hopefully prove useful. Unfortunately, I don't have a macro lens so this is as close as I can get.

I used to wipe away egg cases, and also what appears to be a not so tiny bug (in the center).

Here's the cut off stem where most of the sacs were found. The thin dark lines are pieces of my own hair (I clumsily dropped the stem on my carpet floor).

Here's a full shot of the plant in question, to give more context of its environment. The dotted red line is to show where the stem once stood. The "pot" is a bit odd, I'm sure, but as a college student I have to improvise. It's been sitting on the window of my room since I got it back in November of last year (where it was just the single stem that had to be cut). Horticultural oil sounds like the way to go, as I wouldn't know what to do with a whole swarm of ladybugs in a garden I don't own (I'm renting the room).

Yep. Cottony scale.

Okay, now a question: Any drainholes in your very interesting container? If not, that's stressing the plant.

Further, what did you use as potting mix?

What IS that plant, anyway?