then it does drain the battery ? i mean that is what should be prevented.
There are "rail to rail" op-amps, but I suspect you will find that most (not all) have the same effective protection diodes as logic ICs which clamp to half a Volt either side of the supply rails.
So you are saying that it won't drain the battery but it may damage the op-amp ? You are not being extensively clear here. Anyway, personally i guess i would forfeit the idea that there will not be a drain on the battery at all, but rather go for a very small one using resistors with bigger values rather than trying to cut-out the drain altogether, just make it as small as possible. It would simplify things considerably.
It is (probably) the same as an Arduino. If you power it down, but apply a voltage to an I/O pin, the diodes will power it up to that voltage (less 0.7 V); generally referred to as "phantom powering" and not to be confused with the powering of active microphones in audio systems.
Whether that causes damage will depend on the amount of current drawn and the robustness of the protective diodes.
I understand what you mean, but with an Op-amp i am not convinced that with op-amps this is the case.
They are made up mostly out of transistors and from some of the schematics and references i found, i can not see an issue.
Wiki & Simple on Hackster.
I don't see how any current will be drawn, but i am not an electronics graduate.
Every Op-amp design is different of course, and testing is the only thing i can do to confirm.
Most parts that require power to work are not meant to used 'not powered' , so care needs to be taken in doing that. Still anything would better than to apply 8v to an Attiny pin.
I guess we could just set it up on a breadboard using a LM358 or something.
Have we complemented you on the fine schematics you've posted ? If not, They are great, this is what we like to see !!
It's all in the datasheet.
The LM358: "voltage should not be allowed to go negative by more than 0.3 V
but either or both inputs can go to 32 V without damage (26 V forLM2904),
independent of the magnitude of V+."
Leo..
I am not complaining. You do need to know for the particular op-amp. OK to exceed the positive rail but not below the negative.
Whether "can go to 32 V without damage" means no current draw may of course be another matter. 
Check out Lewis Loflin ....do a bit of research on his site as there are a couple of versions using mosfets in place of power transistor.
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