Field effect transistors
What you describe would usually be a fine way of heating and/or destroying a MOSFET and would not be something one would usually do.
Why are you asking this question?  

You need to say what sort of MOSFET is involved (P or N type).

You need to define what you mean by battery anode. That is less clear than you may think See this article 

Based on hints in your text I'll assume that the battery anode is negative relative to it's cathode and that you have an NChannel FET.
So - the FET Drain is connected to battery +ve and FET source is connected to battery negative. The FET gate is connected to battery positive.

If the battery voltage is below the FET "threshold voltage (Vgs_th) the FET will not turn on, no current will flow and no magic smoke will be generated.

If the battery voltage is significantly greater than Vgs_th the FET will be fully enhanced (turned on). The FET will draw a large amount of current. How large depends on the battery short circuit current capacity, the MOSFET on resistance (Rds_on). If the battery is say 1 12V car battery (lead acid or LiIon or LiFePO4 or ...) then in most cases the MOSFET will be destroyed. It does not make much sense to ask if the p type depletion region will be destroyed. In cases such as this the FET will become a molten smoking blob with no well defined regions retained and all regions essentially destroyed. 

What you describe would usually be a fine way of heating and/or destroying a MOSFET and would not be something one would usually do.
Why are you asking this question?  

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If the FET was a P Channel FET or if the battery was of opposite polarity the FET would not be enhanced BUT the D-S body diode would conduct. If the battery had substantial current capability the MOSFET would be destroyed.
Why are you asking this question?