While there are many gene regulation mechanisms from the cell itself, I was wondering whether it is possible to increase the gene activity in a living cell permanently (so that the protein that it codes for is produced in higher amounts) using methods of biotechnology (i.e. no external factors)? Are there any studies or experiments regarding this topic?
There are technologies available for inducing higher (or lower) gene expression in cultured mammalian/human cells as well as in bacteria. The state of the art is CRISPR-mediated gene activation/repression, where you fuse a transcription activation domain to a Cas gene in a CRISPR-Cas system (usually Cas9), and provide a guide RNA to target genes of interest. This Cas gene is made "nuclease-dead" so that it can only target but not cut or degrade DNA (as opposed to natural CRISPR-Cas systems that do both), and serves to bring the transcription activation domain to your gene of interest, which leads to an increase in transcription of that gene.
It can be done. I've found this article which describes how cou can use a modified CRISPR-Cas9 mechanism to either repress or activate transcription.