In Standard English, this is not a properly formed word; the standard form is "more curious".
As a general principle, the comparative -er suffix attaches to monosyllabic words, and more is preferred with polysyllabic words, so
even though a word like curiouser is readily understood, it has an odd quality to it — this is the reason.
According to the OED , the word curiouser was coined by Lewis Carroll in Alice in Wonderland in 1865, as the phrase "curiouser and curiouser".
In fact, the OED cites this phrase only, and does not treat curiouser as a word unto itself; the phrase has the meaning "increasingly strange".
With this in mind, you might be able to justify the use of the entire phrase in a formal context (depending on your audience).
Addendum:
Let me just state (for the record) that if you are wondering if this would be considered a word in the pure linguistic sense (rather than in Standard English), then I think that most linguists would say yes; it probably has been around long enough and is well-established enough to be considered a word.
english.stackexchange.com/users/449/Kosmonaut
Not quite , apart from in the specific phrase “curiouser and curiouser”, taken from Alice in Wonderland .
Lewis Carroll was playing with language there.
The -er ending is almost never used with words of more than two syllables: so eg more beautiful , more curious , not usually ?
beautifuller , ?
curiouser .
Edit: the OED agrees — it doesn’t list curiouser as a word or a form of curious , but does list curiouser and curiouser as a phrase.
english.stackexchange.com/users/2587/PLL
Curiouser isn't the standard form of the expression in English; the equivalent standard form is more curious .
To elaborate, as a general rule the comparative -er and
the superlative -est suffixes don't attach to adjectives of more than two syllables — the modifiers more and most are preferred pretty much exclusively — so even though a word like curiouser is readily understood, it sounds odd.
This may have been the original point of it.
The OED addresses the phrase curiouser and curiouser within the entry for curious , citing as its origin Lewis Carroll's Alice in Wonderland (1865), which is a notoriously whimsical text.
The OED doesn't treat the bare word curiouser as an expression unto itself; rather, it gives the meaning of the entire phrase curiouser and curiouser as "more and more curious; increasingly strange".
Note that it defines the phrase with the standard form of expression more curious .
But given the entry, you might be able to justify the use of the entire phrase in a formal context, depending on your audience, if not the bare word.
Of course curiouser is a word in the pure linguistic sense, however, and clearly it's a word with its own meaning even as it takes part in Alice's phrase.
And the fact that it doesn't accord with standard English construction and usage doesn't at all stop everyone from using it.
Indeed, along with the entire original phrase, it seems to have entered the collective discourse, if not especially frequently, and still retaining the some of the whimsicality of the original instance.
Witness the following: Mickey thought curiouser was a noun, something that made you more curious.
That's what made up my mind about the title.
Wouldn't it be great if people listened to this album
and it made them curiouser? —
Kate Miller-Heidke Benford’s Law: A curious statistical phenomenon that keeps getting curiouser — Ted Hill
The great thing about couriouser is its self-referential performativity.
It's a strange word that means strange .
Even odder, though, is curiousest , which may help shed some light on the "problem" with couriouser .
I would venture that couriousest , being the couriouser of the two words, is hardly ever used.
Note:
Thanks to Kosmonaut for starting this answer off.
I tried simply to make an edit to Kosmonaut's answer, but my edit was rejected as too different, so I posted the edited version as a separate answer.
Since then I've revised it further.
english.stackexchange.com/users/149428/RJH
It is because Lewis Caroll says it is: “Curiouser and curiouser!”
Cried Alice
(she was so much surprised, that for the moment she quite forgot how to speak good English).
“Now I’m opening out like the largest telescope that ever was!
Good-bye, feet!”
(for when she looked down at her feet they seemed to be almost out of sight, they were getting so far off).
“Oh, my poor little feet, I wonder who will put on your shoes and stockings for you now, dears?
I’m
sure I shan’t be able!
I shall be a great deal too far off to trouble myself about you: you must manage the best way you can—but I must be kind to them,” thought Alice, “or perhaps they won’t walk the way I want to go!
Let me see.
I’ll give them a new pair of boots every Christmas.”
english.stackexchange.com/users/2303/Robusto
Today's description is tomorrow's prescription.
Italian is really really poor Latin (which of course is really really poor something-that-came-before-it), and at the same time each has its rulebook. 'Curiouser' was malformed at the time Lewis Carroll coined it, and that's part of the linguistic humor of it. '
Chortle' was also one of his neologisms and nowadys
I think it is accepted as a 'real word.
As to 'curiouser' I think currently it still sounds strange (because of the (non-prescriptive) inherent English rule that in general multisyllabic adjectives get their comparative via 'more X' rather than 'Xer'.
But if you repeat it enough (and others repeat it enough), it may become more standard.
So in SE standards/culture, using 'curiouser' might be coonsidered a bit too ... informal.
english.stackexchange.com/users/4972/Mitch
Is  curiouser , in fact, a word?
Is “curiouser” in fact a word (like in the famous phrase “curiouser and curiouser”)?
english.stackexchange.com
<meaning><etymology><is-it-a-word><comparatives><register>
english.stackexchange.com/questions/12174
english.stackexchange.com/users/1862/Buns Glazing
In Standard English, "curiouser" is not a properly formed word and the standard form is "more curious". The OED does not list  curiouser  as a word or a form of "curious" , but does list  "curiouser and curiouser" as a phrase. It is not the correct comparative of "curious" which is "more curious" but the word is widely understood. It was coined by Lewis Carroll in Alice in Wonderland, which phrase is cited by the OED. It is suggested that the phrase could be used in a formal setting. 
In Standard English, "curiouser" is not a properly formed word and the standard form is "more curious". This does not follow the rules of standard English, as multisyllabic adjectives normally get their comparative via 'more X' rather than 'Xer'. However, it is possible for words like this to become standard through common usage. The term was coined by Lewis Carroll in Alice in Wonderland, which is a book famous for playing with language. 
 In Standard English, curiouser is not a properly formed word; the standard form is more curious. "Curiouser and curiouser”, is from  Alice in Wonderland .    It is generally considered that "curiouser" is not a word, but has been used in literature.
The phrase is not standard English but a term coined by Lewis Carroll and referenced by the OED. It would be clear to English speakers and could be used. The word "curioser" is a term coined by Carroll.
This does not follow the rules of standard English, as multisyllabic adjectives normally get their comparative via 'more X' rather than 'Xer'. However, it is possible for words like this to become standard through common usage. The term was coined by Lewis Carroll in Alice in Wonderland, which is a book famous for playing with language. 
It is generally considered that "curiouser" is not a word, but has been used in literature.