For sometime I have been trying to come to terms with the concept of hyperreal numbers. It appears that they were invented as an alternative to the $\epsilon-\delta$ definitions to put the processes of calculus on a sound footing.

From what I have read about hyperreal numbers I understand that they are an extension of real number system and include all real numbers and infinitesimals and infinities.

I am wondering if hyperreal numbers are used only as a justification for the use of infinitesimals in calculus or do they serve to have some other applications also (of which I am not aware of)?

Like when we extend our number system from $\mathbb{N}$ to $\mathbb{C}$ at each step there is some deficiency in the existing system which is removed in the next larger system. Thus $\mathbb{Z}$ enables subtraction which is not always possible in $\mathbb{N}$ and $\mathbb{Q}$ enables division which is not always possible in $\mathbb{Z}$. The reasons to go from $\mathbb{Q}$ to $\mathbb{R}$ are non-algebraic in nature. The next step from $\mathbb{R}$ to $\mathbb{C}$ is trivial and is based on need to enable square roots, but since the existing $\mathbb{R}$ is so powerful, the new system of complex numbers exploits this power to create rich field of complex analysis.

Does the system of hyperreal numbers use the existing power of $\mathbb{R}$ to lead to a richer theory (something like the complex analysis I mentioned earlier)? Or does it serve only as an alternative to $\epsilon, \delta$ definitions? In other words what role do the non-real hyperreal numbers play in mathematics?

Since I am novice in this subject of hyperreal numbers, I would want answers which avoid too much symbolism and technicalities and focus on the essence.