Minimum bits needed when representing a value in a set
You can encode the set $X$ using only two bits: use $00$ to encode $1$, $01$ to encode $2$, and $10$ to encode $1e10$.

More generally, the exact identity of the elements of $X$ are not important. You can encode letters of the alphabet or even words, if you want.

You haven't defined what an encoding scheme is, so let me assume that it is a uniquely decodable code. Any uniquely decodable code for $X$ must contain a codeword whose length is at least $\log_2 |X|$ (and so at least $\lceil \log_2 |X| \rceil$). This follows from McMillan's inequality.