Question: Provided that the age of the universe is $13.7$ billion years, but the actual radius is $48$ billion ly, what are we really going to see if we built a telescope powerful enough to reach distances greater than $13.7$ billion ly?
If we look at an object that is $2$ billion ly away, we'll see it as it was $2$ billion years before. If we manage to reach $13.7$ we'll see the universe at its birth.
What happens next?
Due to inflation distances does not correspond directly with time ago. 13.7 billion years ago is actually what you see at the distance of about 48 billion light years away.
What causes this discrepancy is also the same thing that causes redshifts, so things that are more redshifted also have more of a discrepancy between the naive conversion 2bn ly = 2bn yrs ago and the actual relation.
I don't know the exact equation for conversion, but wolfram alpha is able to convert effortlessly if you just put in the distance. E.g; search for 48 billion ly to get 13.7 billion years under lookback time.
You should not confuse the terms light years $[ly]$ to be a unit of time, it is actually a unit of distance, the distance light will travel in vacuum in one year.
When we speak of the age of the universe we use years, in particular about $13.7$ billion years. Now do to the expansion of the universe the distance form which the light comes to our eyes or telescopes can come from region further away then simply $13.7$ billion $ly$. If you do the calculation you obtain a result for the radius of about $45-47$ billion $ly$.
Answering your question, if you look at a distance of about $13.7$ billion $ly$ with your telescope you will simply see a region of space younger than $13.7$ billion years which is nothing special.
Maybe what you are really asking is what we will see if we look for the regions that are the oldest observable one. Well we actually have a picture coming from that period and is the CMB (cosmic microwave background).
This is the oldest light that we can detect, is from when the nucleosynthesis took place, at that time the universe became transparent.
Before that period the only hope to get some information come from neutrino or gravitational waves detection. This is also one of the many reasons because neutrino and gravitational waves detection are so interesting. They can give use new information about the birth or our universe. The only problem is that we have never observed gravitational waves directly (there are proofs of their existence from energy loss from binary systems), and that neutrinos are really hard to detect.
