Napoleon has several ways to stay in power around 1808 up to 1813.
-First thing would be for Nappy to somehow avoid the Peninsula War. Having him support Ferdinand VII's coup against his father Charles IV could do the trick : Joseph Bonaparte wouldn't become King of Spain but Napoleon wouldn't have to use part of his troops to suppress (and fail) into repressing the Spanish rebellion.
-Another scenario involves Napoleon being able to avoid or to win the Russian Campaign. Keeping good relationship with Russia to avoid war would be... problematic. Napoleon and Czar Alexander I didn't get along. You could have the possibility of replacing Alexander I by his brother Constantine, who was an admirer of Napoleon, but I'm not sure it would led the Russians to support the Continental System if it is bad for their economy.
You can also have a different Russian Campaign that ends with a complete Napoleonic Victory if the Russian Commander in chief is not Barclay de Tolly. I'm trying to make a timeline on this subject, but I have trouble in finding the time and I'm still documenting myself not to make any big mistakes.
Barclay was very reluctant with fighting Napoleon directly as he feared (and that could be understood) l'Empereur would crush the Russians in an Austerlitz or Iena-like Battle. Thus, he kept avoiding direct confrontation with Napoleon for which he was heavily criticized, particularly by Bagration. However, Barclay was only replaced later in the campaign by Koutouzov and, by that point, Napoleon was already to deep in Russia to win. Not to mention Koutouzov followed Barclay's policy until the Winter came on and weakened the French army. However, if Barclay isn't the Russian commander in chief, the Russians generals would have had a more aggressive tactic and probably get a beating at Nappy's hands.
-Then you have the 1813 peace offer, where France was to keep only her natural borders : the Netherlands, the Papal States and every land outside of these borders would have returned to their original owners. To reach this, the peace negotiation would have to be successful.
Something similar could happen if there were no Leipzig or if Napoleon was able to somehow crush the Allies earlier.
In any case, the result of Napoleon staying in power would be this :
-A bigger France than OTL, who will have the coal mines of the Rhineland and Belgium. This also means a more industrialized France.
-Britain's hostility. The British didn't like Napoleon and, more importantly, wanted only one thing : that he got out of the picture.
Peabody-Martini said:
It depends on how long Napoleon stays alive without a steady diet of arsenic on St. Helena, who his eventual successor is and most of all he ever able to get his ego under control.
Considering the living conditions Napoleon had on St.Helena and the ages of death of his brothers, I think Napoleon could live at least a decade longer. This would make him die around 1831 or even after that. At that point, Nappy II would be in his 20s if he didn't die.
After that, everything depends on Nappy II. Around 20 is pretty young, but we have younger great rulers in other countries (like Victoria in Great Britain). No one knows what kind of monarch Nappy II could have been, so everything's possible.