"the place you grew up" or "the place you grew up in"

Solution 1:

Both are acceptable and common. In the version with in, the object of a preposition has been relativized; whereas in the version without in an adjunct of location has been relativized.

The place she grew up, they didn't seem to teach the kids much religion, and I'm not one for trying to press my religion on anybody, but Mary was so sort of lost -- that way -- we'd done some talking about it, and she came to church with me sometimes. (NOVEL: Coffin Corner; Dell, Shannon; 1966)

To come back to the place you grew up. Yeah. Now I know. It's like nothing. You find out there's nowhere to hide. (MOVIE: Henry Poole Is Here; 2008)

He choose Saisei to mean West of the Sai-River which was the place he grew up. (WIKIPEDIA)

We have to correct that. Mama, you are talking about giving up our home a place we grew up in. (TV SERIES: Dallas; 1979)

And because I knew that people associated what they saw on the news with the place I grew up in -- a war zone filled with cop killers, killer cops, crack dealers, gang members and lazy welfare mothers -- I convinced myself that embroidering the truth about my living on the Upper Upper Upper West Side was my way of keeping nasty stereotypes of Washington Heights out of people's minds. (NOVEL: Soledad; Cruz, Angie; 2002)

This is the place he grew up in, that fed him and where he finds the articles he sells to survive. (WIKIPEDIA)

In fact, when searching in the iWEB corpus, only 21 out of 76 hits for 'the place I grew up' are followed by in, and most of the rest do not have a locative headed by around or other possible prepositions either.

A google ngrams search puts "the place I grew up." and "the place I grew up in" on about equal footing.

It's pretty well accepted that the relativized element can be equivalent to what would have to be a prepositional phrase if it were not relativized. From The Cambridge Grammar of the English Language p149, on relativized adjuncts:

i

It was a time in my life [when everything seemed to be going right ___]. [time]

ii

I’ve finally found somewhere [where I can work undisturbed ___] [location]

iii

They want to go to the place [where they went last year ___]. [goal]

iv

I shall go back the way [I came ___]. [path]

v

Look at the way [he tackled the job ___]. [manner]

vi

That’s not really the reason [she left him ___ ]. [reason]

The above are mostly understood as prepositional phrases:

i everything seemed to he going right at this time.

ii I can work undisturbed in/at this place.

iii They went to this place last year.

iv I came (from) this way

v He tackled the job (in) this way.

vi She left him for this reason.

The version without in is not, as has been suggested, mainly US usage. Examples of it are common in British English sources as well.

I moved out of my parents’ house very slowly and then all at once. It was the place I grew up — I lived there from the time I was five years old — and the place I always returned to. (The Times)

The place I grew up made me appreciate the importance of a strong community and it provided me with a good local school where the teachers encouraged me to work hard, to think for myself and to be ambitious about what I could achieve. (Fabian Society report)

I also liked the location of the University which is close to London and completely different from the place I grew up. (Canterbury Christ Church University)

Additionally, photographs can create and/or convey memory, this point is very important to me as it reflects the ideas behind my project of documenting the place I grew up. (Max Burton - UK artist)

Solution 2:

You are correct to say that the phrase without "in" is ungrammatical. You don't say "You grew up that place"; it has to be "You grew up in that place". So, out of the two sentences you mention, the only correct one is "What is special about the city you grew up in?".

(You may also say "the place in which you grew up" or "the place where you grew up".)