I am sorry but I don't have enough rep points to comment but am attempting an answer based on my interpretations and having watched all the seasons. Hopefully, it helps.
In regards to Doug Stamper he is one of my favourite characters. Initially, as the first and second season unfold, he is worked extensively and he does out of his own goodwill. He goes above and beyond to help Frank really showing he is comfortable doing the "dirty work". In season 2 when they start to introduce Seth, they are pitted against each other and he tries to out-do him and show his loyalty. There was an episode mid-season 2 where Frank cuts to the camera and talks about "healthy competition" between them.
I personally feel, Doug does it for the power. Yes he is not president but he is the right-hand man. He contributes to Frank and is needed by him as much as Frank thinks he can do it by himself. That is true power. In season 3, Doug over time has collected "dirt" on everyone and has connections, which make him a valuable asset. He refuses Heather Dunbar's insane amount for the key piece of information that would destroy Frank and his campaign (Claire's abortion). The following scenes involve Remy quitting his job and when all else fails...Doug shows up, showing Frank what he was offered and could have done. He wants Frank to know what he is capable of and how much of an asset he is.
Additionally, He does it for his own personal stability as well. Being a recovering alcohol abuser he buried himself in work to distract him from the temptations and going back. He had a slip-up mid-season 2 till beginning of 3. when Rachel got away and Seth seemed like the faithful man. However, when he returned and linking to my first point, he made sure Rachel was dead, he sobered up, and controlled an entire political campaign to an extent behind the scenes. To show Frank
1. He is needed and can do more than what Seth and Remy can
2. He made up for his mistake from letting his emotions getting the best of him and having Rachel escape
Overall, that is my analysis on Doug. I've added some links, which shares similar thoughts. Hope this was enjoyed :)