Chizumatic says (likely permalink):

The future belongs to Robert Rodriguez, who is the only director I've ever heard of who argued for a lower budget for one of his films than the Hollywood studio wanted to give him. Rodriguez believes that tight money brings out the best in a production crew, and his track record backs that up.

See also Serenity, the Firefly movie. I'm the kind of person who enjoys listening to director commentaries, and as I recall the commentary is full of "Well, we didn't have the money to do this, so we had to do that, and I think it came out better" and "Well, the actor accidentally did this, and we didn't have the money to reshoot, so we had to roll with it" and "We told the special effects guys to just come up with something cool within the budget, and they came up with this and it was awesome."

The last one in particular led to one of my favorite special effects scenes in the entire movie, when Serenity falls out of the sky in a catastrophic spiral, something I've actually been wanting to see for a while. (Now I think the only special effect I'm waiting for someone to do is blowing up the Earth correctly.) Complete serendipity, and the money couldn't have possibly been better spent on the usual gigantic ships or explosions or other useless, repetitive same-old-same-old that movies like that usually blow cash on.

Serenity was made for $40 million, and honestly, you'd never really know it; it carries itself like it's worth $80 million.