What is the circuit size and/or circuit depth complexity of an op-amp?
There is no actual convention as to a number of transistors- the designers themselves don't particularly care. What matters is the performance, the manufacturability, the processes used (extra layers, certain types of parts such as good PNP transistors, CMOS, higher voltage-handling ability, and precision resistors require special or extra process steps which add to the cost) and the area of silicon used in that process. A transistor may use 1/10 or 1/100 the area of a resistor so using more transistors and fewer resistors may make for a better design. Capacitors tend to be very expensive and there is typically only one per op-amp. Inductors of reasonable value are more expensive again (or impossible) so they're rarely seen at all. 

Those factors influence the commercial considerations- what the op-amp is worth to the customer (performance) and what it will cost to manufacture (the other factors). 

If you still want to know, you can simply count what parts matter to you for whatever reason (if a schematic is provided in a given data sheet). You'll have to take it with a grain of salt, however, since the schematic may show (say) a differential pair (two transistors) in one particular case, but I happen to know that in that case each transistor is actually something like 100 small transistors in parallel (they're interposed with each other in a tight matrix so as to minimize Vos, TCVos and changes in Vos due to gradients across the chip surface.  

Many modern parts have datasheets with neither a schematic nor a die photo (from which you might be able to suss something out about complexity). You could decap a part and look at it under a microscope or find someone who did, otherwise it's hard to tell.