What are some phenomena that can not be described without the help of Newton's third law of motion?
TL;DR: every time you use momentum conservation.
One way to see this is to take a close look at Newton's cradle:
Image is published under GNU Free Documentation License
You can start with Newton's second law:
$$\mathbf{F}=m\mathbf{a}=m\frac{d\mathbf{V}}{dt}$$
By calculating the scalar product with the velocity vector on both sides of the equation we get the expression for the kinetic energy T:
$$ \mathbf{F}\cdot \mathbf{V} = m \mathbf{V}\cdot\frac{d\mathbf{V}}{dt}= \frac{m}{2} \frac{d(\mathbf{V}\cdot\mathbf{V})}{dt} = \frac{m}{2} \frac{dV^2}{dt}=\frac{dT}{dt}$$
Integration over a path in a conservative scalar field then gives energy conservation. With the potential energy $V$ this looks like $$T_1+V_1=T_2+V_2$$
Just taking energy conservation into account we could get several solutions for different balls moving at different speeds. So something is missing to explain the movement we see. What we still need is conservation of the momentum. This is the point where Newton's third law comes into play. With its help we know that if the ball A impacts on the set of balls B, then they feel a force of $$\vec {F}_{A \to B} = -\vec {F}_{B \to A}$$
With that we can derive momentum conservation:
$$m_A\cdot v_A = -m_B \cdot v_B$$
With momentum conversation the multiple solutions break down to a single one. And what we get is the movement of the cradle.
See also: Does Newton's third law apply to momentum or to forces?
The reason why one often thinks that all the familiar phenomena can be explained just on the basis of the second and the first law of Newton is that it is not clearly emphasized (mainly in school textbooks) that Newton's second law for a system of particles can take the form of $F_\textrm{external}=\dfrac{\mathrm d}{\mathrm dt}(p_\textrm{system})$ only when it takes in Newton's third law. Otherwise it would have been just $F_\textrm{net} = \dfrac{\mathrm d}{\mathrm dt}(p_\textrm{system})$.
The elementary form of Newton's second law is simply $F_\text{net}=\dfrac{\mathrm d}{\mathrm dt}(p_\textrm{particle})$.
When one tries to derive the dynamic law for the total momentum of a system the second law helps to reach $$\frac{\mathrm d}{\mathrm dt}(p_1+p_2+\ldots +p_n) = \frac{\mathrm dp_1}{\mathrm dt}+\frac{\mathrm dp_2}{\mathrm dt}+\ldots +\frac{\mathrm dp_n}{\mathrm dt}=F_\textrm{net1}+F_\textrm{net2}+\ldots+F_\textrm{netn} = {F_\textrm{net}} \;.$$
But the $F_\textrm{net}$ reduces to $F_\textrm{net-external}$ only when one assumes that the forces of particles (of the system) on each other cancels out because they are equal in magnitude and opposite in direction. (Which is what we will call Newton's third law for the moment.) Actually, a stricter form of the third law asserts that these forces also lie along the same line of action and thus helps cancelling out terms of internal torque when one tries deriving the dynamic law for the total angular momentum of a system.
PS: Though it should be kept in mind that neither the weak nor the strong form of Newton's law has unlimited validity. The law clearly comes into big question marks when viewed in the light of relativity of simultaneity. But even in pre-relativistic physics, many simple cases of electrodynamic interactions clearly don't follow any form of Newton's third law.