I came across the problem of ascribing a value to 1/infinity;
There are various ways of doing this depending on what properties you want the extended number system to have.
recently, I heard many things: that it is infinitesimally small
That is one way of doing it, as used in nonstandard analysis. If you take this approach you can do calculus without limits, but at the cost of introducing a huge number of different infinite & infinitesimal numbers. If you are interested, see Keisler's book (free online
(i.e. .0000000000...1 the most intuitive)
That notation won't work! Infinity is not 100000000...0.
that it is 0 (the most ludicrous of them all in my mind),
Well, that is the definition usually used for the complex numbers, where distinguishing between infinity and -infinity doesn't have much use. It is a very good definition for that purpose.
and that it is not definable (which makes the most sense, although is a bit of a let down).
It's not definable if you want to have all the axioms of arithmetic still holding for infinity. Neither is infinity itself. If one wanted to be difficult one could say that 1/infinity is defined but infinity isn't!