I think that might be a different mindset I have often observed between my EU ans my US friends; US friends keep talking about competition and becoming market leader etc, my EU friends just want to run long term companies. I know many people here who run sites for 15-20 years that make enough every year since inception for a comfortable (meaning building pension and saving) 3-4 people company. Even with free offshoots in the domain that were already there sometimes when they started.

You definitely can compete with huge companies that have no or automated support; you probably cannot beat them, but you can sustain a good profitable company for a long time and they might buy you in the end if they see you grow in your region while they do not.

Competing with free is challenging, but maybe there are other examples to choose from. Snapchat doesn't cost money for consumers to use it, same as Facebook. That's not due to competition with Facebook, and arguably it's not competing with a Facebook offshoot either. There are a number of analytics platforms depending on what you're looking for (eg Mixpanel and New Relic) that come to mind readily. eBay faces competition from a little company called Amazon.