I’ve coined this term, because I’ve never heard of this concept, but there’s probably another word for it. I’m referring to words as “abcde…,” in which “a” is a word, “ab” is a word, “abc” is a word, etc. Obviously, in English, these words would have to begin with letters that can be words in and of themselves.
Is there an existing name for these words?
What is the longest such word in the English language?
Just in case someone requires an example, I think you’re talking about this phenomenon:
He
Her
Hero
Heroin
Heroine
Not sure there will be a term for this, as it’s interesting, but not very useful (in an everyday sense)
To be clear, are you referring to a series such as:
a
an
and
Andy
Or specifically a set of words in any language such as that it literally follows the standard lexicographic order of its alphabet?
ETA: Nevermind, I assume Mangetout is probably right
I did a Google search, found nothing. The longest one of those I could think of was '‘noted’.
Is ‘n’ a word though?
‘Amides’ is a six letter example.
There’s an entry for it in the dictionary. Some people seem to think that makes it a word.
I checked Borgmann’s Language on Vacation and he calls them curtailments, obviously looking at it as removing letters, but it comes out to the same thing. His examples are pasterns, reversers and albertines. Looking up any unfamiliar words is an exercise for the reader.
I haven’t checked Word Ways yet; it’s too late at night. Perhaps tomorrow.
“Would you like to save your file? y | n > n”
Granted that’s an abbreviation, but if we want to be generous I’d say you could consider exceedingly common abbreviations as words themselves.
I think he’s interested in those sequences where you add only one letter, not two, to the end of the previous word.
Sure. Every single letter is a word referring to the letter, or the letter’s form. And some of them have additional meanings as well. For example, Merriam–Webster defines “n” as (among other things) “an indefinite number; especially : a constant integer or a variable taking on integral values”.
Not sure. One category is a superset of the other anyway.
Word Ways calls them curtailments too. Here’s an example from a 1976 issue.
If you’re not going to care about how many letters you add to the end of the word, then the exercise of finding the longest such word is trivial: it is simply the longest word in the English language, and the sequence of accumulations contains (at minimum) its first letter, and the entire word.
Under your definition, a more interesting problem than the one posed by the OP is to find the longest sequence, rather than the longest word.
I think they can only start with A, I, or O. Best I can come up with is
I
id
ide (Collins English Dictionary)
idea
ideal
ideals
or
a
as
ass
asse (Random House Dictionary)
asses / asset
assess / assets
Here’s one I found years ago, but have never mentioned to anyone. Unfortunately, it uses an abbreviation, albeit a very common one:
c
co (=company)
cos (lettuce)
cost
costa
costar
costard
costards
All the words can be found on M-W online
Why? Isn’t C the speed of light as well as a commonly used computer language? Why can’t it be used?
psychonaut, are you a subscriber to Word Ways, too?
Yes. Beginning with one letter, then adding one more at a time.
And I’d consider A, I and O to be the only true 1-letter words. The others are more like symbols (not implying that all words aren’t symbols).
I used to be, but I had to let my subscription lapse for lack of time for reading. Maybe in the future I will resubscribe. 
Agreed- it isn’t interesting at all if there isn’t a series of more than two words.
I almost did the same, but then I found that if I put the latest issue in the bathroom, I’d get it read while doing other business.
Those who consider A, I and O as the only one letter words have a rather limited idea of words. Most dictionaries have every letter listed as a noun in addition to any symbolic meanings and abbreviations. And you can’t tell me C (as in the programming language) is either a symbol or abbreviation.
C, as in the programming language, is a proper noun. Generally in word games of this type proper nouns are not allowed. c, when used to mean the speed of light for example, is listed under “symbols” in dictionaries that I have looked at. So not everything listed in a dictionary is necessarily a word.
You may be on firmer ground with the idea that C is the name of the letter C itself, but if it counts as a noun at all I’d consider it a proper noun.
If we limit ourselves to “familiar” words (i.e., words that you might hear in conversation), there is:
a
an
ant
ants
antsy