Asking which are the biggest “self-publishing” companies in the world is like asking “What are the 10 biggest casinos in Vegas?” In short, these are the companies that are best at helping themselves and opportunistic profiteering from authors’ hopes and dreams. Since Amazon is the creator of this modern paradigm (e.g. publishers are greedy and outdated and you don’t need one anyway), let’s start with them. Sure, some people that you could count on both hands and feet have achieved financial independence through the Amazon bookshelf model. Ignoring the fact that the vast majority of the remaining author pool does not earn enough to cover their expenses—let alone their time—Amazon continues to devour other corporations, newspapers, chain retailers, digital content groups, and any substantial competitor or partner. Their owner is literally the richest man in the world. So the biggest self-publishing company is the one that you create with financial independence, meaning it does not rely on the mechanisms and distribution of global corporations to exist.
There are a few companies who will push you to “sign a contract” for a book/book chapter/whatever, which you should not do as they are fraudulent and will make you look bad. We usually negotiate for the advance and book royalties after we have the manuscript signed. There are no “agents”, you have to do all that part yourself. Do not trust anything your publisher tells you, they might be lying to you. So what do you do? First get a copy of the manuscript. Then get copies of any marketing material. Then read the book/chapter before you start it, as it is your responsibility to learn from all your mistakes. I suggest starting by opening the chapter, looking to see the structure and feel of the work, then taking a moment to read the main text and see if you think that you have the idea..