What changes the number of outstanding shares?

other than
- a split
- a public offer for capital raise
...?

Shares issued to executives DOES NOT adjust the shares outstanding, as these SHARES MUST be registered with the SEC, just like a normal public offering. So, if they are issuing more stocks through a new offering, than this would, of course, also adjust the shares outstanding. But this is obvious.
There are three other measures that can adjust this figure: the Issuance of Convertible Preferred Stocks and/or the Issuance of Convertible Bonds; and finally the Issuance of Warrants (as these are convertible into commonshares). While these do not immediately adjust the shares outstanding, it can very easily adjust them if they are converted...which is why shareholders are required to vote on these.
Please also bear in mind that shares repurchased by the Company through a buyback (which the shares then become treasury stock) are NOT considered shares outstanding. So this will affect the amount as well.

hope this helped.

The company may issue stock options to executives. When they execute those options then there are more outstanding shares. Companies also sometimes buy back shares in an effort to boost share prices. They may also make an aquisition by issuing shares.

Exercise of stock options by employees and excercise of share warrants issued at time to share sale to raise capital

Buybacks by the company.