The phrase "big leagues" means the highest level or most important competition in a particular field or sport. It suggests being at the top of one's game or in a highly competitive and prestigious environment. Full definition
With moves like these, though small, we can expect to see other bigger clubs in bigger leagues who make transfers of millions of euros to also try out cryptocurrencies. (crypto-lines.com)
And I believe this is one of the real secrets of how to be a standout big league pitcher. (si.com)
Start with what an invite to big league camp really means. (sbnation.com)