Orbital Hyperloop for Approach to $c$? We have a bunch of "orbital catapult" ideas to get things into orbit - most involve some sort of loop with mag levitation in vacuum. There is a practical limit on the top speed here due to whatever is being shot having to go through the atmosphere. 
What if we built this loop in orbit for imparting initial velocity on a spacecraft / payload? (Other than removing the concern of burning up through the atmosphere)
What limits the speed of an object in a hyperloop? What keeps us from accelerating the object close to c?
Edit: Unrelated to bounty, but the comments got me thinking - put an object in orbit, beam energy to it to get it faster and faster while it's still within a certain proximity to earth, then let it escape. I am guessing this would waste a lot of energy to stay in orbit at higher speeds though.
 A: Launch loops should be attached to a planet, which supplies practically infinite reaction mass and tensile strength, though limited angular momentum (mass times velocity times radius), necessary for high circular orbits around a planet.  A large asteroid can be a reaction mass anchor, but won't directly supply tensile strength.  Rotating tethers is the best bet for smaller bodies.
Energy is cheap; the Sun emits 380 trillion terawatts.  Some of that can be beamed from space solar power satellites (SSPS) to rectennas powering launch loops;  indeed, the beam polarity can be rotated to cross-polarized rectennas, directly powering the launch loop drive motors.  Beam energy to vehicle kinetic energy conversion efficiency will be rectenna-limited;  if that is 80%, then 10 terawatts of beam energy can launch 150 tonnes per second to a constellation of construction ports, more than 4 billion tonnes per year.  That power can be generated with less than 10 million tonnes of SSPS.  
Given the current global launch market of less than 3000 tonnes per year, that won't happen for a while, but it's nice to know there is room for growth.  
Slingshot around the Moon is one way to add angular momentum to a loop-launched vehicle, but the Moon is in the proper position for this only twice a month, while a loop can launch thousands of vehicles per day.
An amusing kludge is to launch some of the vehicles to a very high apogee, fire a small rocket, and add lots of angular momentum up there (small velocity times huge radius).  If the perigee of the resulting orbit intersects the apogee of a direct launch from Earth, you can arrange to exchange the momentum there, with rotating tethers or an Arnold/Kingsbury spaceport.  See http://launchloop.com/ConstructionPort for work in progress.
Want to go somewhere at the speed of light?  Digitize yourself and send the information to a distant receiver.  The atoms you are made of are as common as stardust, and tend to disperse in less than a century.  The receiver can be sent as a small replicator, along with software to build the receiver, at relatively slow speeds.  This is an enormous technical challenge, but tiny compared to moving anything sizeable at at relativistic speeds (CERN LHC moves nanograms of protons for gigabucks - scale from that).   Humans are, after all, the end product of tiny replicators.
