As we know, light travels with a constant speed and the Universe is ever-expanding and endless. Considering this fact, is it possible that the sky we see should change everyday? Let's say today we see 100 stars in the sky because the light from those 100 have only reached Earth. There could be distant stars whose light has yet to reach Earth, so we might see another 100 stars the very next day.
You're right that there are numerous stars that are so far away and were created so "recently" that their light has still not arrived at Earth. However, most processes in the Universe happen on extremely long timescales (for human standards). In the Milky Way, on average approximately one star is born every year (see e.g. here). But the timescales from the initial collapse of a molecular cloud and until ignition of the star are of the order $10^5$--$10^6$ years (see e.g. here). This means that new stars don't simply "pop up" on the sky, but instead slowly evolve.
The theoretical limit to what we can see is given by the distance that light has traveled since the Big Bang. The Universe is 13.8 billion years old, but due to the expansion, we can actually see 46.5 billion lightyears away. As we also look back in time the farther we look away, we see the Universe at this distance before it had formed any galaxies at all, though. So, a hundred new stars cannot be seen every day. Instead, every molecular cloud, protostar, main sequence star, old star, and many other objects are seen every day as a tiny bit more evolved than the previous day, where "a tiny bit" means "not at all" for most practical purposes.
Note, however, that some processes do take place on very short timescales. For instance, when a massive dies, it suddenly explodes in a supernova. Here you see a clear evolution from day to day. The even more energetic gamma-ray burst evolve on timescales of seconds to minutes, and neutron stars spin at rates of several hundred times per second.
Thus, if for instance you instead of stars ask about supernovae, then the answer would be "Yes, the light from distant supernovae has yet to reach us, and approximately once per day a new supernova pops up one the sky."