Can endergonic reactions occur outside of living organisms? If the Gibbs free energy equation is defined as:
∆G = ∆H - T∆S
And the amount of energy/work released from a reaction is:
-∆G = w_max
Living organisms use exergonic reactions to metabolize fuel where the Gibbs free energy is negative and they (can) occur spontaneously in non-living structures.
Exergonic: ∆G<0
Also in living organisms the energy that is released from the exergonic reactions can be coupled with the endergonic reactions. These do not occur unless work is applied to the reagents.
Endergonic: ∆G>0 
Given that, is it possible for an endergonic reaction to happen anywhere outside of a living creature or man-made creation?
 A: Yes. Endergonic reactions are happening, for instance, in any place where water is naturally freezing into ice. In the surface of a lake for instance.
As you state, any endergonic process happens always coupled with an exergonic (the atmosphere air above the lake is pumping enthalpy away from the lake). But, if you consider the lake together with the nearby atmosphere layer as the system, the process is exergonic. That happens too with living things.
You find that some processes are endergonic because you are isolating a piece of the system. When you add the environment, only exergonic processes can happen.
A: The Gibbs Free energy is not defined as ∆G = ∆H - T∆S.  That holds only at constant temperature.  It can be defined as $\Delta G = \Delta H - \Delta (TS)$.  For a reaction to be spontaneous at constant temperature and pressure, $\Delta G$ should be negative.  However, all reactions go to a certain extent, even if only microscopically.  Basically a $\Delta G$ of zero means that the equilibrium constant for that reaction is 1.  A negative $\Delta G$ means that the equilibrium constant is greater than one, while a positive $\Delta G$ means that the equilibrium constant is less than one.  This isn't magic, the relationship is $\Delta G = - RT\ln K$, where $R$ is the gas constant, $T$ the temperature, and $K$ the equilibrium constant.
One way to have a reaction take place to a significant extent with $\Delta G > 0$ is to couple it to another reaction with a large negative $\Delta G$.  This is most often done by having one component in the desired reaction also a component in the "driving" reaction.  And this can be done without applying work to the the reactants.
A: Yes. For example: triple-alpha capture during helium burning in stars.
