Would it be possible for someone, once flung, to continually orbit a small planetoid? I am wondering this question (though it may seem obvious) because of an issue I'm having with a peer. He says that if a person is flung around a small planetoid, at just the right angle and velocity, they would be forever moving. I say that this just simply isn't possible. Am I missing something, or is he?
His argument word for word "There is a center of gravity and as the character gets further away from the center, it slows down as it starts to get pulled back in, but the gravity pulling it in speeds it up and overshadows the gravitational pull, eventually starts to slow down again and resets the cycle."
 A: If by "flung", you mean "launched from the surface", then it would be very difficult.
Unless you've got some sort of thruster to change your orbit, to the first approximation your path that started on the surface is going to once again strike the surface.  You can't jump with your legs and go into a near-circular orbit around the body.
Of course planetoids aren't perfect spheres and the irregular gravitational field would allow some modifications to the orbit.  But those are generally small perturbations of the orbit, not big changes.
There is no problem with someone once in orbit to remain for a long time.  The earth is still orbiting the sun and we don't expect that to stop any time soon.  As there are no significant drains on the energy of the system; it just stays in place.  This isn't a concern for perpetual motion.
For a person orbiting a planetoid, the orbit might be stable for several years.  But the lumpiness of the planetoid and the tugs from other planets in the solar system will eventually cause the orbit to decay.  It will either intersect the planetoid and crash or be flung from the body entirely (where the object would now be in a simple solar orbit similar to the planetoid).
A: As pointed out by contributor BowlOfRed: orbital motion is inherently cyclic. That means: in the idealized case an orbit repeats itself; a closed loop. In the idealized case: an object starting its orbit at a particular point will after a full cycle go once  again through that very same point.
From here on I will refer to the celestial object that will be orbited as 'the primary'.
Therefore, if the orbital motion starts at the surface of the primary then the orbital motion will return to the surface.
Moreover: if at the start the launched object moves at an angle to the surface of the primary then the corresponding orbit is along a trajectory that intersects the surface. That means: the launched object is destined to impact the primary.
Insertion into long duration orbit requires two acts of change of velocity: the first is the launch that brings the object up to orbital height, the second change of velocity is circularization of the circumnavigating motion.
Without that circularization the circumnavigating motion is in a very real sense not orbital motion; without circularization the launched object is destined to impact the primary.

As mentioned by BowlOfRed: in the not idealized case there is the fact that the gravitational field of the primary will have lumps, and generally those deviations from perfect spherically symmetric gravitational field tend to cause change of orbital motion. You get cycles where the shape of the orbit cycles between more circular and more elongated.
Example, there are not many lunar satellites because this effect tends to make orbit around the Moon unstable, even when it starts with good circularization. For more information see the answer to the space.stackexchange question: Which artifial satellites in lunar orbit are currently active?
