Construct an infinite number(?) that has a beginning, an infinite middle, and a end; such as 1000...0001, or 98111...1114 etc. Has this type of number been explored? Under some simple multiplications, 5(1000...0001)=5000...0005, other mathematical operations are not determinable.
1/1000...001, or 1/5200...0008, etc. may have different infinitesimal properties. Can the surreal numbers include these?
π(1000...000) would be sort of like a specific infinite ω and π(1000...000)/(1000...000)=π. Surreal number types as in πω/ω.
There is a way of writing surreal numbers called "Gonshor's sign expansion". Basically, every Surreal number is a "string" of +s and -s (actually a map from an ordinal to {+,-}). For the finite strings, this matches somewhat closely to tally marks and binary. ""=0, "+"=1, "++"=2, "+++"=3, "-"=-1, "--"=-2, etc. $``+-"=\frac12=.1_2$, $``\underline{++}+-\underline{++-+---+-}''=\underline{10}\,\,. \underline{110100010}\,1_2$, etc.
However, there are infinite ordinals like $\omega$, which give rise to numbers like "+-+-+-+-..."=2/3, but those numbers have no end. Luckily, lots of ordinals do have an end, like $\omega + 3$. Then you get numbers like "+-+-+-+-... +++", which is probably 2/3+3*"+-------...", where "+-------..." is a positive surreal less than "+-"=1/2,"+--"=1/4,"+---"=1/8, etc.
I don't know if this is satisfying to you, but it is a number system where some of the numbers have infinite representations with ends.