“Every corporation organized under this chapter shall have such officers with such titles and duties as shall be stated in the bylaws or in a resolution of the board of directors which is not inconsistent with the bylaws and as may be necessary to enable it to sign instruments and stock certificates which comply with §§ 103(a)(2) and 158 of this title. One of the officers shall have the duty to record the proceedings of the meetings of the stockholders and directors in a book to be kept for that purpose. Any number of offices may be held by the same person unless the certificate of incorporation or bylaws otherwise provide.” [Section 142(a) of the Delaware General Corporation Law]
What this means is that you should have at least a President and Secretary. However, all of our startups start with a President, Secretary and Chief Financial Officer. The Chief Financial Officer or Treasurer will be required if you plan to qualify to do business most other states. Also, I think it is generally good practice to have one officer position dedicated to the financials and accounting of the company.
Some starts will also add a Chief Technology Officer or any other officers. However, simple is best for startups in the interest of efficiency and moving quickly.