Hard to choose, but I'll go with
Scarecrow
(1973). It appears that Hackman and Pacino both gave it their stamp of approval, too. From the IMDb Trivia page:
This film is often overlooked in both lead actors' canon, but it is often cited by both Al Pacino and Gene Hackman as one of their favorite films.
Gene Hackman has stated that his performance in
Scarecrow
(1973) is his personal favorite.
According to Al Pacino, this was the best script he'd ever read.
Gene Hackman was asked in a rare interview about his favorite movie of all the ones he did. He said it's really hard to pick; it's just like picking one of your children. However, he said, " …there was a movie I did years ago; didn't get much press; it was called
Scarecrow
."

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The French Connection
Unforgiven
The Conversation
I have to add The Heist, No Way Out and Zandy's Bride as well. I saw Zandy's Bride only once on TV, but it left quite an impression on me. The only performance of Gene Hackman that I didn't care for was in the Superman series. I loved the films, and he wasn't bad in them, but I just didn't think he was right for the part of Luthor.
edit:
and Scarecrows, of course.
Never be complete.

Oh that's tough to have to choose just one, but I'll have to go with
Mississippi Burning
.
It is my business to protect your majesty. against all things.

Hoosiers.
What we got here is failure to communicate!

1. Bat 21
2. Company Business
3. Target
4. Class Action
5. Narrow Margin
6. Loose Cannons
7. The Package

The Conversation

Extreme Measures

BONNIE AND CLYDE (my fave film ever)

Hard to choose just one cuz he made so many great films but if it has to be just one, I guess it would be: Hoosiers.
I also really liked him in.
Superman, Crimson Tide, Mississipi Burning & one I rarely ever see mentioned..the underrated 1990 thriller: Narrow Margin.
"You're not a doctor!" "You don't have enough medical background to squeeze a pimple!"

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The Conversation
His best performance IMHO!

I reevaluated my thoughts and I have to go with Mississippi Burning. However, I still love him in Hoosiers, however I think his performance and dialogue in MB are slightly better than in Hoosiers. I have seen both films so many times and it is because of Hackman. I wish he still was active, but hey he deserves to retire.
"Yeah well, you know, that's just like your opinion man."

mississippi burning, the french connection, unforgiven

I loved him in "Mississippi Burning" and "The Poseidon Adventure".
Koleniko, John Belushi and Corey Johnson fan for life.

I don't think Hackman has ever given a bad or awkward performance on film. My personal favorites are Hoosiers and The Conversation. I choose favorite films based on whether I sit down and watch part of it when it shows up on TV, even if I've seen the film a dozen times before (and I've certainly seen Hoosiers that often). Hackman is also superb in supporting actor roles in films like The Firm, The Runaway Jury, No Way Out, Bonnie and Clyde and even The Birdcage. The man is simply believable in every role he plays and that authenticity is the mark of a really good actor.

Hoosiers.
Obamunism: The end of the Republic.
At times like these, I really hate being right!

Probably Crimson Tide, or Run Away Jury.
Unforgiven was probably his best performance though, movie was just too long though.
"What's the most you've ever lost in a coin toss?"

"I Never Sang for My Father," followed by "The Conversation." This man is fantastic in everything, one of the great understated talents.

The French Connection..

Mississippi Burning
"You must have a really wide foot, you got both of them"

Bonnie and Clyde
French Connection
Mississippi Burning
Crimson Tide
Enemy of the State
As you can see I love him in action roles

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The Conversation definitely, there are few films that are as good as this one!