Can a manager fire someone without the CEO's knowledge or permission?

It depends on the size and policies of the company and the position of the terminated employee.

A “Chief Executive Officer” is a specific title that often reports to a Board of Directors. In some companies, the CEO may be required to “sign off” on all terminations so there is the image of transparency related to personnel actions in the company but insulation for the members of the Board should that termination result in civil action. In most cases, though, the CEO is just a signature acknowledging and approving the manager’s and HR’s recommendations. If anyone is going to be sued over it, the “buck stops” with the CEO.

In other companies, the CEO may only be involved in authorizing the termination of executives and direct reporting staff - the Senior Vice-Presidents, VPs, Directors of… - with the input of HR and Legal. If the CEO was involved in recruiting or selecting an individual deeper in the organization, they may be informed and consulted at the end of the investigation for their recommendations on if or how to “bend” policy regarding the employee’s level of discipline.

If the termination will have significant operational impact, like deadlines won’t be met or a project will have to be postponed or even ended, then the CEO (and maybe the CIO, COO, CFO, too) will be consulted for next steps.

In my experience, HR and the immediate leadership of a position (its supervisor and that supervisor’s supervisor) will carry out terminations independently from anyone on the executive team or CEO.