A LT1637 is advertised as 1.1MHz, 0.4V/us Over-The-Top Micropower, Rail-To-Rail Input and Output Op Amp. This features are suitable for current sensing application since input voltages are right at the upper rail. The circuit show current sensing of 5V supply system, but  monitoring up to 44V application can be done by this circuit  (the maximum supply voltage of the op-amp). Here is the schematic diagram of the circuit:

current sensing circuit schematic

Besides the complication of extra parts, the VOS performance of op amps at the supply is generally not factory trimmed, thus less accurate than other solutions. A small source of gain error  is introduced by the finite current gain of the bipolar transistor. [Schematic diagram source: Linear Technology Application Notes]