I know that any rational number can be expressed as a continued fraction, but what about irrational numbers? For example, what is the continued fractional representation of Pi, or e for that matter? Can all real numbers be expressed as continued fractions? For example, can the following number be expressed as a continued fraction?
0.1234567891011121314...
All real numbers can be expressed (not necessarily uniquely) as continued fractions. Check out the Wikipedia article on the subject, in particular from this point on down for the special examples you requested.