Today, 14:03
Yes! You have it. Exactly: if you have no further information ("no theory" in your words) you can ask 2 questions, hoping that you will find the number quicker, but if you are unlucky, there is a higher chance that you will need to ask 4 questions. So 3 is the overall optimal: it's always 3, no risk. There is no way to solve it with 2 question without risking. If you have some information ("theory"), that usually it's AB, and rarely it's CDEFGH, you can go for AB first. But in this case the symbols are not equiprobable so it's not random. And we are going for random content, so you don't have more information what the result usually is (what distribution the symbols usually follow).
If we do the game with 16 letters, it's the same: the optimal is 4 questions. With 32 it's 5. No matter how big your range is, the optimal way is always log2(n).
Going even higher doesn't change the game: If you have 4096x8 random bits, your optimal way to find out those bits is to do it in 4096x8 questions. Can't do it in less.
Practical compression example: try compressing random files with any compression software. The result will always be a bit bigger. In case of paq* it will actually try finding patters (find a "theory" behind the bits and bytes). It really does try finding patterns! Being a random file we know that there are no patterns there. But paq does not know that. So it tries.
When it believes it found some theory (that some symbols are more probably than others) then it applies it - and experiences a loss. Ouch! So the result gets a bit bigger. Like trying to guess the 8-symbol game in 2 question. Risky right? Because it is programmed to lower the risk and try finding the best theory it will give up soon trying to solve the 8-symbol game in 2 questions - it will stick with 3 questions. And so it reduces it's loss. So this is the reason why there is always a small loss there - because it is always doing a bit riskier at the beginning, so those losses are inevitable (in case of paq or any other compression software with similar internal workings).