Why won't a 1.5V (or 1.2V) battery give an electric shock?
The current a voltage source (like a battery in this case) delivers is up to the load.  The current rating is only the maximum the source can deliver.  It can't somehow force this onto the load.  For example, if you connect a 1.5 k&Omega; resistor accross the 1.5 V battery, only 1 mA will flow.  This will be true whether the battery or power supply was rated for 1 A, 10 A, or 100 mA, since all those are more than the 1 mA the load will draw.

What causes harm to your body and what you feel is current, not voltage.  The published levels of how much current is safe vary quite a bit out there depending on what part of the body and what point the author is trying to make.  You want to be extra careful running current accross the heart and thru the brain.  Between two fingers of the same hand is relavtively safe in that it will be uncomfortable before you get hurt.  You can safely put your fingers accross a 1.5 V battery because the skin resistance prevents any significant current from flowing.  As shown above, it would take 1.5 k&Omega; just to get 1 mA.  Normal skin resistance is in the 10s of k&Omega; at least.  You can try dipping your fingers in salt water and then touching both ends of the 1.5 V battery and see if you feel anything.  Again, make sure this is the fingers of the same hand.  That way just in case you get any significant current it will stay local to the hand and not get to the heart or brain.  You can definitely feel a 9 V battery on the tounge, but I don't know if even wet fingers is enough to feel a 1.5 V battery.  If you try it, let us know.