てしょうがない, is a common colloquial expression used when ‘(A) cannot be helped’, or something is ‘extremely (A)’. This structure is simply an abbreviated version of 仕様（しよう）, a noun meaning ‘way’ or ‘method’, as しょう is easier to say than しよう. Literally, the meaning is close to ‘it’s (A), and nothing can be done about it’. Generally, this just implies that the extent of (A) is so high that it cannot be overlooked, or cannot be avoided.
This structure may be seen following the て-form of verbs, い-Adjectives, of the で-from of な-Adjectives.

• 日本語（にほんご）が上手（じょうず）になりたくてしょうがない。
  I really want to become good at English.
• 部屋（へや）が暑（あつ）くてしょうがない。
  My room is extremely hot.
• スイーツが好（す）きでしょうがない。
  I love sweets so much that it can’t be helped.

Alternatively, 仕方（しかた）がない may be used, but there is an extremely small difference between these two structures, with the noun 仕方（しかた） also meaning ‘way’ or ‘method’. Traditionally, 仕方（しかた）がない is perceived to be slightly more polite, so may be more likely to be found in literature.

• 私（わたし）は今（いま）妊娠（にんしん）をしているが、寿司（すし）を食（た）べたくて仕方（しかた）がない。
  Although I am pregnant right now, I really want to eat sushi.
• 眠（ねむ）くて仕方（しかた）がなかったので、少（すこ）しだけ昼寝（ひるね）をしました。
  Since I was extremely sleepy, I took a small nap.
• この古（ふる）いパソコンは不便（ふべん）で仕方（しかた）がないから捨（す）てることにします。
  Since this old computer is extremely useless, I will throw it away.