The computer geek's biggest dream was to hack his way into the Defense Department's mainframe

[nohabloingleas]

The computer geek's biggest dream was to hack his way into the Defense Department's mainframe

Why can't you use the pronoun "his"?

 

[teechar]

Who told you that you can't use "his"? Perhaps that geek was a female.
Note that I have also changed the title of your thread. Please do not use "u" as a short form of "you".

 

[SoothingDave]

Who said you can't?

 

[emsr2d2]

Note that I have changed your thread title. Teechar had already corrected the "u" to the correct word "you". I have changed your title so that it is the sentence you are asking us about. That's what titles must always be. Your actual question should appear only in the main body of your post. Whoever told you that "his" is wrong was wrong! It's fine as long as the computer geek is male. If they're female, use "her". If you don't know the gender, use "their". I'm sure it wasn't a native speaker who told you it was wrong.

 

[nohabloingleas]

Pronouns like his and he should not refer to a possessive noun like geek's.

Second exercise in https://chompchomp.com/hotpotatoes/proref03.htm

 

[emsr2d2]

Pronouns like his and he should not refer to a possessive noun like geek's.

The question was the second exercise in at/from/on https://chompchomp.com/hotpotatoes/proref03.htm

Remember to write complete sentences on the forum.

 

[SoothingDave]

It's suggesting that you re-write "the geek's dream" to "the dream of the geek."

Which is unnatural. There is nothing wrong with the original sentence.