Can we determine the probability of finding an electron without a potential well? Can we determine the probability of finding electron in a region of space if there is no boundary condition?
For example, if the wall of a one-dimensional potential well is infinitely far away then is there a way that we can find the probability of finding the electron in a given region of space?
 A: I would rephrase the correct answer by Thatpotatoisaspy   as: 
when there are no boundary conditions there are the plane wave solutions of the quantum mechanical equations. Plane waves go from -infinity to + infinity so they are not good for modeling the probability of  finding the electrons in our experiment. But, there exist wave packets, which can  model the probabily:

The calculation tools for computing the probabilities of interaction of elementary particles do not need such modeling as they use the Feynman diagrams of interactions, which have specific rules that write down the integrals from the diagrams, without a  need for modeling individual incoming particles, and they are quite good in their predictions for the probability of interactions.
A: I think it's important to point out that if a problem admits bound states, then there is always a boundary condition at infinity, which comes from demanding that the wave function go to zero as $x \to \pm \infty$.
It's interesting to note that for a particle in a box, the quantisation condition that leads to discrete solutions comes from demanding that the wave function vanish at (say) $x=0$ and $x=L$. Similarly, in problems like the harmonic oscillator or the hydrogen atom, the quantisation conditions arise by imposing the condition that the wave functions vanish at $\pm \infty$. (This is worked out in Griffiths' text.)
Now, the case that you mention in your question is essentially that of a free particle (a particle in a box with the walls infinitely far away), and as mentioned in the answer by Thatpotatoisaspy, you can construct a wave packet to define the particle as being ''localised'' somewhere, and thus you can speak of a particle as having some probability of being found somewhere in space.
A: Yes. A good example is a wave packet for the free particle. The ordinary free particle solution is not normalisable, but you can construct another solution which is, the wave packet. 
