The following sentence was written by a Japanese.
Is 食べてたい just a typo, or is it a short form of something, like 食べておきたい or 食べていただきたい?
“食べてたい” is a colloquial and contracted form of - ”食べていたい - I want to keep eating.”
Likewise, “寝てたい – I want to stay in bed,” “起きてたい ‐ I want to stay up (all night),” and “(一晩中)喋ってたい - I want to keep chatting (all night)” are used in place of “寝ていたい,” “起きていたい,” and “喋っていたい.”
“…てたい” is colloquially spoken by both young and older people today, but to me it sounds like a childish turn of phrase. I don’t recommend you use this expression in the written form or in formal occasions.
As noted in the comment by strawberry jam, this is most likely a contraction of 食べていたい "want to be eating", the desiderative (wanting) form of 食べている "to be eating". In informal speech, it isn't uncommon for the い in various -ている forms to be contracted out: 行ってます, 話してた, 読んでる, etc.
The contracted form of -ておく is -とく, where the て + お becomes と. Conjugated variations include 積んどいて, 置いといた, 片付けときます, etc. The IME on my Mac recognizes these -ている and -ておく contractions, and I think I remember that the IME under Windows does too.
I'm not aware of any contracted forms for -ていただく (though I suspect such forms probably exist, perhaps in dialect).