What is the fastest speed a spaceship can achieve?

There is no theoretical "maximum" speed for a spacecraft. Any distance can be traversed nearly instantaneously with an https://www.wikiwand.com/en/Alcubierre_driveThe limiting factor is power. With current tech, it takes an impossibly large amount of energy to power an Alcubierre drive. However, it's only a matter of time before we figure out how to either make the drive and/or power source more efficient.

There are so many limitations on the reality of spaceship speeds within our three dimensional space, that we would have to ignore them to be the fastest.  We need to step outside our spatial world to get anywhere beyond our current technology capabilities, i.e. deep space.  Then instantaneous transfers of a spacecraft through space would be the fastest. The question then becomes, could instantaneous transfers ever become a reality?  According to some, it is already a reality; probably NASA already knows it.In backing up that statement, I reference a speech given by a very reputable scientist who was a director of many top-secret innovations for the military ‘black projects’; Ben Rich making statements that we have the means to reach the stars and the capability to send ET’s back home. Mr. Rich’s speech was given almost a quarter of a century ago to group, all alma mater engineers from UCLA. There are many web links to this speech, but one stands out based upon a question ask on the propulsion system technology as he was heading for the door leaving.His reply was another question, “Do you know how ESP works?”  The response was “I don’t know, all points in space and time are connected?”  Mr. Rich replied, “That’s exactly how it works” and he walked out the door.What Mr. Rich was alluding to was not some paranormal process to send spacecrafts across spatial distances instantaneously, but instead since all points in space are connected; it is for that reason allows instant transfers. Transferring objects from point A to point B instantaneously is teleportation.  The missing ingredient is the science to perform it, and ‘black projects’ has that also.  The web link:http://www.openminds.tv/lockheed-skunk-works-director-says-esp-is-the-key-to-interstellar-travel-video-1092/23042http://www.openminds.tv/lockheed-skunk-works-director-says-esp-is-the-key-to-interstellar-travel-video-1092/23042