It may sound strange to some people, but I think the game Medieval II Total War has some great immersion, if you enjoy the time period and can imagine a thing or two about warfare. I can just imagine me giving out the orders, and ruling my empire with an iron fist.
Right now, playing as the Egyptians, me (the Sultan, of course), and the Crown Prince (who strangely enough is older than me, and probably wants to kill me, but that chivalrous sun of a ***** can't lose his reputation over a simple squabble like this) are out on campaign. We just vanquished the Byzantine Empire, when we just heard word of a new threat coming from the east.
Having strengthened our positions in the West, we are returning to the Holy Land, where we will amass the horde and fight this threat, in the name of Allah! Of course, only after our numerous spies have infiltrated their armies, and our dreaded assassin's have slain their leaders. Ah, I almost forgot, I should send out a diplomat, perhaps they are willing to surrender! Muhahahahaha!!!
Not many people can find a game like that immersive I reckon. You're looking at a few stylised icons of units, and a map of Europe with a bit of Africa and the Middle East depicting your empire. Yet the background of all the units, the individuality of your generals, the strategically sound ways of fighting, and the historical phenomena happening around you are extremely well done I think.