One of the major problems I have with hard-line prescriptivists is that they follow their convictions to the point of absurdity, arguing that something completely standard ought to be changed because it doesn’t conform to a rule they’ve decided is inviolable. Today’s example is aren’t I.
Yes, I has a problem. Well, it’s not so much a problem with I, but with its companion am. Unlike the other conjugated forms of to be, am doesn’t form a contraction with not. Are and is are flexible, contracting equally readily with a pronoun (we’re) or the negation (isn’t). But am apparently fancies itself too good to consort with a debased negation. And so we find a hole in the English language, a word that should exist but doesn’t: amn’t.
Unlike am, English as a whole is flexible, and so another word (aren’t) pulls overtime and fills the hole. And this earns the ire of the accountants of the English language, who fume and fuss that this isn’t in the job description of aren’t. Didn’t they negotiate an agreement between subjects and verbs that aren’t can work with you and we and they and other plural subjects, but not with I?
So there is a hole in English, and there is a word that fills it. But filling the hole requires breaking a common rule in English. What do you do? If you are like pretty much every speaker of English, you break that rule. But there are those who put rules above reasonability and consider aren’t I bad grammar. Let’s look into the matter.
History. Aren’t is first attested in the Oxford English Dictionary in 1794. Google Books offers examples from 1726 and 1740. All of these are instances with you or they as the subject. As for aren’t I:
(1a) Aren’t I rich? You know I am! Aren’t I handsome? Look at me. 
(1b) “I’ve got threepence,” she said, “Aren’t I lucky?” 
(1c) “Aren’t I?” seems to be thought the correct thing; but why should we say “Aren’t I” any more than “I are not”? 
Aren’t I appears in Google Books by the 1870s, and writing is conservative with respect to spoken usage, so aren’t I likely appeared in speech much earlier. In the earliest attestation — (1c) from 1872 — aren’t I was already perceived as standard. No one still alive today spoke pre-aren’t-I English. So if it’s been standard for 130 years, why wouldn’t it be fine still? Here are some possible (but misguided) objections to it.
Logic? The primary objection to aren’t I is that it has subject-verb disagreement. You wouldn’t say I aren’t, so you can’t say aren’t I. The first part of that is correct, but the second doesn’t follow. After all, if I aren’t being incorrect blocks aren’t I, why doesn’t are not you being incorrect block aren’t you?
You can’t apply simple logic to language and expect there to be no exceptions. Emily Morgan has noted before that the logic of language is far more complex than prescriptivists make it out to be.
Informality? One site claims that aren’t I is unacceptable in formal writing. But that’s the case for all contractions, not just aren’t I, because they’re informal transcriptions of speech. The fact that aren’t I doesn’t appear in formal writing is no more a condemnation of it than the fact that aren’t you doesn’t appear in formal writing. (And, by the way, both do appear in formal writing.)
Alternatives. Now, let’s say you’re unconvinced that we should leave well enough alone, and you really want to fix aren’t I. How are you going to do it? Look at the prominent alternatives that are available for aren’t I: am I not, amn’t I, ain’t I. Am I not is fine if you’re being poetic or intensely formal or need to stress the negation, but in most cases, it’s going to sound completely unnatural and overly stuffy. Amn’t I is perfectly fine if you are Irish or Scottish, where it persists as a standard form, but it’s exceedingly rare outside of those Englishes, and you’ll look affected if you use it in another dialect. Furthermore, it’s hard to pronounce the neighboring m and n distinctly, so people may think you’re using ain’t I instead. Ain’t I, of course, used to be a standard form, and Fowler himself fought in its favor, but nowadays is one of the most condemned words in the English language, one that will make even most moderate prescriptivists write you off as ill-bred.
The fact of the matter is that there is no other option that is acceptable in most English dialects and at an appropriate formality level. This is why aren’t I has taken hold.
Suppletion & Syncretism. I want to conclude with two final reasons why aren’t I shouldn’t concern you: suppletion & syncretism. Suppletion is a specific type of irregularity, where one irregular form fills in (or overtakes) the regular form. Usually, suppletion is talking about a case where the irregular form is from an unrelated paradigm: e.g., better instead of gooder in English, or mejor instead of más bueno in Spanish. No one complains that better is wrong because gooder follows the rules better. With aren’t I, the suppletive form is only from a different part of the paradigm, not a whole different paradigm, but the basic idea is the same. There is a seemingly regular rule (add n’t to the conjugated verb) that in one instance is ignored in favor of an irregular form. If you want aren’t I done away with, you ought to want to see better consigned to the scrap heap as well.
Furthermore, it’s only suppletion from a contemporary perspective. Actually, we’re dealing with syncretism, where two distinct syntactic forms happen to look identical. David Crystal has a very nice explanation of the history behind aren’t I, which came from people mistaking an’t for aren’t in non-rhotic (“silent-r“) dialects. Genealogically, the aren’t in aren’t I and the aren’t in aren’t you aren’t the same. Which means that, technically speaking, aren’t I isn’t an example of subject-verb disagreement; it’s a case of mistaken identity of one aren’t for another.
Summary: No, aren’t I isn’t incorrect. It’s been in use for at least 130 years, the alternatives are all insufficient, and the “logical” arguments against it are fallacious. It’s no more incorrect than using better instead of gooder.