The CRISPR technique for splicing genes into organisms has been applied for a few years now, often quite successfully. It's been applied to at least 3 babies in China for genes linked to increased IQ, along with countless experiments on non-human animals.
There are many open questions as to the safety of the technique. CRISPR appears to increase the mutational load of an organism, scrambling a few adjacent genes during the splicing process. There is also the possibility of accidentally including sections of bacterial DNA also, a byproduct of the techniques' origin – hijacking a bacterial defense process evolved to combat bacteriophages.
CRISPR-Cas can do more than genes, however. Recent experiments illustrate that it's possible to splice in entirely new chromosomes (collections of hundreds or thousands of genes). These could even be induced to code for amino acids that are beyond the alphabet of natural Earth genetics.
Human beings typically have 23 pairs of chromosomes, though some creatures have many more. Chickens have 39 pairs, Hermit Crabs 126, and some ferns have as many as 630. The number doesn't seem to correlate with the complexity of the organism, more the opposite, perhaps.
It should definitely to possible to splice in some extra data to our DNA if we wish. Perhaps we could encode a genetic memory that includes language or technical capabilities, enabling our descendants to access master skills instinctively, out of the box. We might even carry certain digital processes within the cells as a method of proving that we are a living being, the ultimate CAPTCHA.
For those looking for a laymans term explainer on what CRISPR actually is, here is a pod cast episode explaining exactly that. : https://samharris.org/podcasts/humanity-2-0/
A few extra articles for those looking for a deeper dive.
https://www.nature.com/news/synthetic-double-helix-faithfully-stores-shakespeare-s-sonnets-1.12279