Whats the effect on Voltage and Amps when 244 9v batteries are connected in a series?
Assuming the batteries are identical, the current won't be more than one battery could supply into a short circuit (probably several amperes for a fresh 9V alkaline battery- I've measured 5A briefly from a Costco 9V battery) no matter the number of batteries. The voltage increases with each added cell, but so does the internal resistance, and by the same factor. 

That said, there's potential for explosion if the internal resistance of one started to increase greatly (which it might from such abuse). Also, a couple amperes at 2.2kV is a lot of power, liberating a lot of heat even if the batteries stay balanced and they're closely packed so there's nowhere for the heat to go. 

It is also easily a lethal shock hazard. 

And putting that kind of voltage with that much available current into a Radio Shack handheld meter is reckless. 

NiCd 9V batteries can supply 90A briefly which could well lead to a fairly substantial explosion- 90A at 2.2kV is going to create a stubbornly persistent arc, so it might not be just a single battery going up. 

I remember reading some speculation (in an old issue of Analog, if I recall correctly) about an office-desk-size arrangement of AA batteries producing something like 1MV. The author speculated that it would be next to impossible to extinguish the arc once initiated. 

The only limit on this insanity would be the geometry of the arrangement- if the voltage per cm gets high enough between any two points to break the air down you're going to have instant issues. But you could make a linear arrangement of 150,000 batteries and get 1.35MV (peak available output power about 3MW) if you could round up that many batteries- 4500 HP- almost 7 metric tons of batteries and still not quite what you can get from an internal combustion engine in a drag racer.