I'm using NOP Playground 3.3 with no other 3rd party plugins or code.
I'm on a shared hosting plan.
I'm having app pool restarted several times and then stopped.
My hosting company, Arvixe is telling me this is due to either the code or my website is hitting the memory limit (600MB)
Do you have any calculation of memory consumption of the Playground theme and if the 600MB limit is really not sufficient to run it.
Also is there any way that I can know for sure what the real reason behind this app pool restart?
The memory consumption does not depend on the theme but on the data (products/categories) you have in your store. Please read this post about memory consumption in nopCommerce.
The new version of nopCommerce 3.4 will have a lot of performance and memory improvements and it might need less memory but if you want to be sure you won't have any problems my advice is to not use a shared hosting. Here is a thread in the nopCommerce forum for the new improved version that will be released in a couple of weeks.