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For as long as I can remember, my teachers have taught me never to use first person in an essay. Yet, some of the examples we read in classes, or older writings we have to annotate, are written in first person. So what is the actual rule here? Why is using first person in essays considered unacceptable?

This depends on your style guide and potentially your teacher/school/boss/etc.'s guidelines. If your teacher (for example) says to avoid using the first person you may be able to negotiate to change their position, but ultimately they are the ones evaluating your work so you need to follow their rules.

None of the three major American style guides forbids using first person.

This is what the APA Blog has to say about the first person:

I am often asked why APA Style prohibits the use of I or we. I love this question, because the answer is always a pleasant surprise: I or we is perfectly acceptable in APA Style! In fact, the Publication Manual actually recommends using first person, when appropriate, to avoid ambiguity.

(Another APA blog post also addresses this issue.)

According to a Q&A on the CMOS website, the CMOS says this:

Please see CMOS 5.220, s.v. I; me: “When you need the first person, use it. It’s not immodest to use it; it’s superstitious not to.”

Another Q&A on their site says:

Avoiding the first person used to be considered proper, but now it’s considered very formal, if not old-fashioned. It’s not a question of correctness, however; both styles are correct. If you feel strongly that the first person is out of place in your work, don’t use it.

MLA, on their webpage, says:

Let the first-person singular be, instead, a tool that you take out when you think it’s needed and that you leave in the toolbox when you think it’s not.

My recommendation is to use first-person pronouns only for attribution. If you do something original, such as a thesis or other research, it makes sense to use "we" for things you did with your supervisor, or "I" for what you managed on your own. It's important to examiners/reviewers to know what is one, what is the other and what is instead due to previous academics, and to be confident of your non-plagiaristic honesty on that front. Laurel's answer shows a number of style guides for such writing gives similar advice.

If your essay contains nothing original, such as when you give an overview of the arguments for and against philosophers' views on an issue, the fact you wrote the essay is immaterial, and you should effectively be as anonymous as a Wikipedia article's author(s). Although you don't go into detail about where you read an author using "I", my best guess is it was when they made a case for something of their own.

Like with so many other issues of style, it depends on your audience.

For a school essay, your audience is your teacher. So write the essay your teacher wants, no matter if it's okay to do it another way elsewhere.

If you submit an essay or article for publication, the audience is the publisher, and you should follow its rules. A publisher curates for a larger audience and will have rules and expectations for authors that fit those needs.

Some academic essays need to be in first person. Imagine a college application essay written in 3rd person. I'm sure people have done it. I wouldn't advise it. If you're writing an essay to read out loud at a conference or other presentation, and it's about your work, you also need to use first person or it's just weird and confusing.

Figure out your audience and adjust your writing accordingly.