I am wondering if the ku/katta conjugation for i adjectives has (had) a
meaning by itself.

For example tanoshii becomes tanoshiku. In the past tense then tanoshikatta.
Clearly the tta is the indicator for the past tense. So my question is: is or
was the ku/katta for i adjectives a verb ? Where does it come from ? And what
is its meaning (beside the one learned in textbooks). My japanese teacher also
wondered but she mentioned that ka, ki, ku, ke, ko were used in old japanese
to indicate past or so. I have heard once that i adjectives have the desu/da
already built in so maybe ku/katta is some variation of da/datta?
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i-adjectives
I adjective ku/katta
Japanese-as-a-second-language learners generally remember 楽しかった and so on as a
form of 楽しい. However, in the "traditional" Japanese grammar, た/だ is indeed a
standalone word categorized as a 助動詞 (auxiliary verb).

  * [Auxiliary verbs in Japanese](https://japanese.stackexchange.com/q/63351/5010)
  * [Why Japanese verb has so many forms?](https://japanese.stackexchange.com/q/61902/5010)

So yes, た in 見た, た in 簡単だった, and た in 楽しかった are exactly the same 助動詞 for the
past tense. For example,
[this](https://dictionary.goo.ne.jp/word/%E3%81%9F/#jn-132183) is the entry
for た in a monolingual dictionary, and it clearly says it's an auxiliary verb
that signifies the past tense.

く and かっ are considered as part of the "true" conjugation forms of
i-adjectives (namely [連用形](https://japanese.stackexchange.com/a/65953/5010)).
They are not regarded as a word, and they don't have any intrinsic meaning on
their own. The following table is the conjugation chart of i-adjectives
described in Japanese Wikipedia.

[![形容詞の活用表](https://i.stack.imgur.com/1J3sG.png)](https://i.stack.imgur.com/1J3sG.png)
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Short answer: we may never know how -く arose. -かった (e.g. 楽しかった) can be
synchronically analyzed as the contraction of -く and あった (i.e. 楽しく + あった).

Long answer:

The origin of -く is indeed tricky, since as
[naruto](https://japanese.stackexchange.com/users/5010/naruto) mentioned in
the comment it is attested from antiquity.

According to Frellesvig, B. (2010). _A history of the Japanese language_.
Cambridge: Cambridge University Press.

> we interpret these formants as forms of a restricted copula ('adjectival
> copula') and gloss examples accordingly, e.g. _samu-ku_ cold-ACOP.INF
> 'coldly'. This interpretation of the formants attaching to the adjective
> stem finds further support in its relations to some other grammatical forms

and considers the resemblance of the adjectival copula to the past tense
auxiliaries ("ki ... used in old japanese to indicate past or so") in your
question.

As for -katta, it is more straightforward.

> In OJ (=Old Japanese) the existential verb _ar-_ was used with the
> infinitive of the adjectival copula ( _-ku_ ) , the regular copula ( _ni_ ,
> _to_ ) and the negative auxiliary ( _-zu_ ) to form analytic forms, and
> these combinations sometimes fused phonologically: ADJ- _ku ar_ - => ADJ-
> _kar_ -, NOUN- _ni ar_ - => NOUN- _nar_ -, NOUN- _to ar_ - => NOUN- _tar_ -,
> VERB- _zu ar_ - => VERB- _zar_ -. In EMJ (=Early Middle Japanese) the fused
> forms gave rise to secondary conjugations

It may not be historically (i.e. diachronically) accurate to say that 楽しく and
あった contracted to form 楽しかった, since the form あった is attested much later than
the phonologically fused form たのしかr-. However, just like it is perfectly
acceptable to say that the word "ability" is from "able" + "-ty" (rather than
tracing it all back to Latin words _habilis_ and _habilitās_ ), the synchronic
analysis of 楽しかった as 楽しく + あった is a perfectly valid one.

For completeness, other analyses of forms historically related to the
existential verb ある include:

  * 楽しかろう, synchronically 楽しく + あろう
  * 堂々たる, synchronically (and diachronically) 堂々と + ある
  * 許されざる, which is 許されず + ある
  * 華麗なる, which is 華麗に + ある
  * 良かれ悪【あ】しかれ, which is 良く + あれ + 悪【あ】しく + あれ

Most of these forms are now quite dated though.
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