Ideally, what sort of characteristics should a transistor have for a regular linear amplifier?
from: http://www.ti.com/lit/an/sboa117a/sboa117a.pdf

Meet the transconductor (aka operational transconductance amplifier), which has the following (alas unphysical) Vce/Ic characteristic (forgive the awful mspaint job).



Available as the OPA861. Sort of.

EDIT:

When you learn about BJTs, you learn the hybrid-pi small signal model:







simulate this circuit &ndash; Schematic created using CircuitLab

This is the large signal model of the transconductor.