Why would anyone implement delta-gamma neutral hedging? I understand you wouldn't lose money, but conversely, you wouldn't make money either. Like if I buy 1 XYZ 50 call option w/ a delta of, say, 20, and I short 20 shares of XYZ to become delta neutral. Next day, the call's delta raises to 30, so I short an additional 10 shares. Day after that, the delta falls to 10, so I cover 20 shares. What's the purpose of this? Can anyone explain this to me?
Your example is delta neutral, not delta-gamma neutral. What you describe, well more or less, is called gamma scalping, use the search function to find numerous discussions on this subject.
An example of Delta and Gamma neutral would be a box; Short 50 Puts Long 50 Calls Short 55 Calls Long 55 Puts Theory says you should make the risk free interest rate when long a box, as above. In practice however, with spreads and commissions to pay, it's somewhat different.
I haven't been very profitable at it so, just a little better than b/e so I'm not much of an expert, but imho it is a way to capture volatility without much risk (overnight holds are not a problem), except maybe bleeding to death.