For 5 TL, you can visit the huge, impressive, and completely breathtaking (literally; wait until you try those stairs) Rumeli Hisari in north Istanbul. I saw a photo of this on a photo blog I follow and knew I had to make it a part of my trip to Istanbul. I am so glad I did; it was one of the highlights of my visit to Turkey.
If you have trouble with mobility or are very afraid of heights, you may get limited enjoyment out of parts of the fortress. If you're just a little nervous about the idea of ascending lots of unrailed narrow stone steps--know that it's worth it for the views at the top! I couldn't believe how much of the fortress we could walk to, climb on, and see, and also couldn't believe how few people were visiting. I guess it's far enough from Sultanahmet that many don't make the trip. That said, I encourage you to visit if you can. You will absolutely not regret it!