"I waited for him for half an hour." You must have "for" in 2 places in that sentence.

I wouldn't say "I waited for him half an hour", however I might say "I waited half an hour for him". It just sounds incomplete to say "I waited for him half an hour", like something's missing.

I waited for him half an hour is quite correct. It is not essential to add in a second "for". So look at this example: We waited (for) hours to get the tickets. However when we use a direct object after wait, we have to use "wait for", so in this example below we add in "for": "Wait for us outside the cinema. We’ll be there at seven o’clock". For more information and other usages of what to insert after wait or can't wait please book a lesson with me.