I think one depressing example is innovation in weapons and other dangerous fields. "If we don't build it, someone else will first" is unfortunately historically been shown to be true, has it not?
Today's unsavory borderline reactionary doomposting brought to you by: my crippling fear that I'm isolating myself in a political echo-chamber (so naturally I gotta hop online and exclusively ask my fellow leftists)
The one that I get is "competition breeds innovation". I think they're just wrong that it should always be monetary competition. Personal-prestige competition (such as academic reputations) can also be a powerful motivator. The USSR had a bit of this going on in the various OKBs.
Having dabbled in academia I have witnessed firsthand that no amount of economic/business rivalry will ever measure up to the unhinged competition of fanatically curious nerds