What would happen if a company bought back of all of its stock? Would it own 100% of itself? Companies occasionally own some of their own stock. A company owning its own stock is actually pretty weird when you think about it. If I own 1% of a business's stock (and therefore own 1% of the business's assets), and the company owns 50% of its own stock, I as a stockholder of the business, indirectly control 0.5% of the business. Through the 0.5%, I own 0.25%. Through the 0.25%, I own 0.125%, and ad infinitum. This adds up to 2% (which makes sense, since if I own 1% out of 50% of the stock, I would own 2% of the business.)

What happens though if a company controls all of its stock. This would mean that the company owns itself. In particular, it isn't really owned by any individuals anymore. The purpose of a public company is to create value for its shareholders, but what does that mean if it owns itself?

Practically, how would this affect things like voting and bankruptcy?
The New York Stock Exchange has a continued listing requirement of 300 shareholders and 200,000 shares. A company will want to consider whether continuing to be traded on a stock exchange is desirable before approaching these limits. 

LINK to a NYSE reference document.

If a company buys back almost all of its shares the last individual or institutional owner (who ever owns the last outstanding share yet to be bought back) will will own all of the equity. This person or institution will control the company. The seller of the last outstanding share will want the highest possible price. If this trade must be done via the company buying it's own shares, because that is the premise of your question, the owner will direct the company to pay the maximum cash amount. The company will generate the maximum cash by liquidating the company. 

Definition from Wikipedia... 


  Liquidation is the process in accounting by which a company is brought
  to an end


LINK to Wikipedia article about liquidation.

If the last outstanding share can trade for more than liquidation value, there is no reason to liquidate the company. The owner can sell the share to a different investor in order to get the highest possible price.