[i-adj 〜い → 〜くて] + たまらない [na-adj 〜な → 〜で] + たまらない [verb て-form] + たまらない
The grammar pattern ~てたまらない expresses that a feeling, sensation, or desire is so intense that one can't bear it. It conveys that the speaker is overwhelmed by an emotion, physical sensation, or urge to the point where it feels unbearable or uncontrollable. It is often translated as "extremely," "unbearably," "can't help but," or "dying to."
This pattern is used exclusively to describe the speaker's own feelings or sensations (or to ask about the listener's). It cannot be used to describe a third person's feelings directly, unless reporting what they said.
When combined with the たい form of a verb, it expresses an overwhelming desire – "dying to do something."
This pattern is used with involuntary feelings or sensations – things you can't control. It would sound unnatural with deliberate actions. For example, you wouldn't say ×勉強してたまらない, but you can say べんきょう勉強したくてたまらない.
~てしょうがない and ~てしかたがない have very similar meanings and are often interchangeable. However, ~てたまらない tends to convey a slightly stronger or more visceral sense of being overwhelmed.