Do humans today still carry DNA from the very earliest humans?

The answer to this question might not be as straightforward and simple as it first appears. The reason for this is because the question is somewhat vague. The definition of the "very earliest humans" is up for interpretation. Does that refer to Neanderthals, Denisovans, or something earlier or later? If Neanderthals are sufficient, then the answer to the question is yes: modern humans share some of that ancient DNA but not much of it. According to recent research, Neanderthals have contributed somewhere between 1% and 4% of the DNA of humans of Eurasian descent. The reason for the Eurasian descent part is because Neanderthals did not live in the ancient African area; therefore, no interbreeding occurred between these populations. We are fortunate to have enough surviving DNA to make these comparisons, and scientists have discovered that similarities exist in both mitochondrial DNA and nuclear DNA. Some of the comparisons being made are actually gene-to-gene comparisons as well.
http://humanorigins.si.edu/evidence/genetics/ancient-dna-and-neanderthals

http://humanorigins.si.edu/evidence/genetics/ancient-dna-and-neanderthals/interbreeding

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4988579/