〜てたまらない・〜てしょうがない
unbearably, terribly, dying to
い-adj くて · な-adj で · Verb て-form + たまらない / しょうがない / しかたがない
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てたまらない says a feeling or sensation is so strong you can't stand it: 暑くてたまらない, "it's unbearably hot". たまる means "to bear", so it's literally "can't bear it".
It's used for your own feelings and physical sensations: hot, cold, sleepy, painful, worried, lonely, and wanting something (会いたくてたまらない, "dying to see them").
てしょうがない and てしかたがない mean the same: "it can't be helped, it's so…". しょうがない is the more casual of the two.
い-adjectives use the くて form, な-adjectives take で (心配でたまらない), and verbs use the て-form. Don't use them for other people's feelings without そう or らしい.
Examples
暑くてたまらない。
It's unbearably hot.
国の家族に会いたくてたまらない。
I'm dying to see my family back home.
試験の結果が心配でたまらない。
I'm terribly worried about my exam results.
足が痛くてしょうがない。
My feet are killing me.
朝から眠くてしかたがない。
I've been terribly sleepy since this morning.