What does 老得 means in here? Does it mean
I'm too old and have to bother colleagues constantly.
I find an example,
I'm too old to have a cat.
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You should take 老得 separately in this case.
老: always; constantly
得(dei3): need; have to
我老得麻烦同事 means I have to bother colleagues constantly.
In 我老得不能养猫了(I'm too old to have a cat), 得 is pronounced as de and 老 means old.
It relies on context to determine whether 得 is pronounced de or dei3, which reflects different meanings.