I was told 眠たい was an adjective, and that it is an exceptional case. How does it work as an adjective and what is its relation to the verb 眠る?

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'眠たい' is a colloquial version of '眠い', 'sleepy.' Maybe it has something to do with '眠りたい'(want to sleep), no? (Hmm... maybe not) –  Choko Jan 22 '12 at 1:28
What do you mean by "an exceptional case" –  fefe Jan 22 '12 at 1:59
@fefe in that it is related to a verb. –  yadokari Jan 22 '12 at 5:00
There is a somewhat similar case with 煙たい. (Conjugations can be described as 'irregular'. I'm not sure if it's useful to talk of derivations as 'irregular'. –  Bathrobe Jan 24 '12 at 0:11
@Bathrobe san, Ah, 煙い and 煙たい. 眠い and 眠たい. Then 重い and 重たい, right? What else... –  Choko Jan 27 '12 at 14:51

Yes, 眠たい is an adjective. It's a synonym of 眠い, meaning "sleepy." (大辞泉 entry here.)

From what I can find (specifically, 大辞林's explanation), 眠たい is in fact related to 眠る, but the 「〜たい」 here is different from the desiderative "I want to" 「〜たい」. This 「〜たい」 is a shortened form of 甚{いた}い (or いたし, in its original form in older Japanese), which is a 連用形 verb ending in classical Japanese meaning "extremely X, where X is the state related to the verb." Here, this "state" is sleepiness. Note that this 甚{いた}い can attach to verbs and give them an emphatic adjectival meaning, and does not attach to adjectives.

めでたい (happy, fortunate) has a similar etymology: 愛{め}で (from the 連用形 of 愛{め}ず, the old form of 愛する) ＋　甚{いた}し.

Edit: Just to clarify, the original form of 眠たい was 眠{ねぶ}り甚{いた}し, but a combination of phonetic drift and other factors led it to eventually evolve into its current form.

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It is an i-adjective because it conjugates as so.

Its relation to the verb 眠る cannot be handled systematically within derivational morphology. The stem of 眠る is nemur-, whereas 眠たい only uses a portion (nemu-) of it. So it has to be considered that this formation is idiosyncratic, not systematic. Note that the meaning is also idiosyncratic.

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「眠くない」は、「眠たくない」では・・？（「眠たい、眠たかった、眠たくない」と「眠い、眠かった、眠くない」がセットになると・・） –  Choko Jan 22 '12 at 4:01
@Chocolate Thanks. That was my mistake. –  user458 Jan 22 '12 at 4:09