Every black tea bag contains around 25-110 mg of caffeine per serving. If you want more caffeine and tannins, of course 5 tea bags would contain more caffeine than a single bag. The reasons are:
When you use a tea bag, caffeine, tannins diffuse out the tea leaves into the water. The longer you leave the tea bag/leaves in the water, the closer to equilibrium you will get. But if one tea bag, there would be limited amount of caffeine, tannins.
Besides, five tea bags would mean caffeine, tannins diffuse in parallel in the hot water. Diffusion rate depends on concentration as well as temperature of water as this paper shows.
The diffusion of these caffeine, tannins will be fastest in the first few minutes, where the concentration gradient between water (no caffeine, tannins) and the tea leaves (full of caffeine, tannins) are the steepest, as well as highest temperature.
While after optimum time, the slower the diffusion of caffeine and tannins into the water, because the concentration of the water(or maybe now tea) and the tea leaves are similar, and lower temperature. ie. the amount of caffeine, tannins that could be diffuse out the teabag is not much after long.
But if you add hot water to the tea bag again, you would find the teabag has flavor because hot temperature and steepest concentration happens again.