Possible to make full loop on a swing only by "pumping legs"? Is it possible to make a full $360^{\circ}$ loop on the type of swing in a children's playground if I power the swing only by "pumping" my legs? i.e. nobody pushes me?
 A: Let's solve this in two parts. First, I will calculate how fast you need to go at the bottom of the swing in order to be able to make a complete loop (without the rope going slack). Next, I will estimate whether you can achieve this speed by "pumping" - that is, moving your center of gravity around to increase your speed.
Part 1: speed needed
The velocity you need can be determined from the length of the swing. If the rope is length $\ell$, you need to go fast enough that you are able to resist the force of gravity at the top of the loop. 
At the top of the loop, you have lost some speed due to gravitational potential: $\Delta E = 2 m g \ell$. And in order to have sufficient acceleration to counter gravity, you need a speed to keep the rope taut:
$$\frac{m v_{top}^2}{\ell} \gt m\;g$$
This means that the swing needs to move with a minimum velocity (at the bottom) of
$$v_{bottom} \gt \sqrt{g\;\ell + 4\;g\;\ell} = \sqrt{5\;g\;\ell}$$ 
Note - this does mean that you will feel significant g forces at the bottom of the swing... Don't try this with your toddler!
Part 2: doing it under your own steam
In principle this only depends on the friction forces on the swing, and how much energy you can add per "pump". However, I think in practice the answer will be "no" - and here is why:
The action of "pumping" on the swing is basically an application of conservation of angular momentum. By moving your center of mass closer to the center of rotation, you increase your velocity. There is a very nice explanation of this in this answer. The key on a swing is that you "pump" near the bottom of the swing, when it is going fast; and you "release" when you get to the end of the swing (where your velocity goes to zero). Normally, you move just a short distance, and add just a little bit of energy. But to make a full loop, you need to add sufficient energy in a single "pump" to go all the way around.
So let's assume you were making a 180° swing when you decided to try to add the additional angular momentum. This is the most you can swing without the rope going slack (unless you do a full 360). In this case, the velocity at the bottom of the arc is
$$v_{init} = \sqrt{2\;g\;\ell}$$
Now you need to increase your speed in a single "pump" to $\sqrt{5\;g\;\ell}$ to reach the velocity calculated above - but in effect you are changing $\ell$ when you pump, so we would need to take that into account. 
You can do this if you can move your center of mass closer to the center of rotation. If you stand on the swing and push yourself up, you might move the center of gravity from $\ell$ to $\ell'$. If you continue standing, you will be making a tighter circle which will help at the top of the swing.
In this scenario, the velocity (you standing) at the bottom of the swing $v_b'$ has to be sufficient to get you to move up by $2\ell'$ - in other words, we need
$$v_b' = \sqrt{5 \;g \;\ell'}$$
and we know from conservation of angular momentum that $$v_{init} \ell = v_b' \ell'$$
Now we can solve for $\ell'$. A bit of manipulation shows
$$\ell'^3 < \frac25 \ell^3\\
\ell' < 0.74 \ell$$
This means that if you can raise your center of mass by about 25 % of the distance from the seat of the swing to the pivot, you could do a loop.
Assuming you are a 180 cm person with 80 cm legs, you can raise your center of mass (from crouch to standing up) by about 40 cm. If that is 25% of the distance to the pivot, you would need a very short swing - rope length no more than 160 cm. You might hit your head... and the force you need to do this is very significant (you are lifting more than twice your weight). If you can't do a deep squat with a friend sitting on your shoulders, you will not be able to do this.
I think the answer is "maybe; but don't try it at home..."
