Do eggs reach a point of no longer being safe eventually?

Eventually, sure, but you'll want to get rid of them long before that. Not only do they gradually develop a bad flavor (particularly if they're stored near strong-smelling foods like onions), but they start to lose their functional qualities such as the ability to thicken sauces and make baked goods rise. The main danger of keeping eggs too long is a loss of quality and taste, so if you’re not a regular egg-eater, don’t buy of the 60-egg pack at Costco, despite the irresistibly low price.
 
When eggs go “bad,” they usually give off a sulphureous odor as the protein breaks down, says food scientist Susan Brewer. They should be thrown out at that point.
 
Source(s):
University of Maine - Cooperative Extension
Facts About Eggs
 
Susan Brewer, Ph.D., University of Illinois, Department of Food Science and Human Nutrition