I have had a request come through to increase the pool size in IIS which I assume to be the application pool size.
I have attempted to search for more information regarding what the pool size is, what it does, its importance as well as caveats in increasing its size.
Am unsure where to find it in IIS 7 and 7.5 as well as what the default size is.
Does changing the pool size also affect web gardens?
As noted in the comments, the question is ambiguous.
There is a Private Memory Limit (KB), which defines how large an application pool can grow in system RAM before it gets automatically recycled. For example, when you open Task Manager (taskmgr.exe), you'll see an instance of w3wp.exe running for each application pool. Add the Working Set column. I believe that corresponds to the Private Memory Limit setting.
There is a CPU Limit which dictates how much CPU an application pool is permitted to consume over a period of time. Its default is zero, meaning no limit.
There is a Queue Length setting which dictates how many simultaneous requests can be serviced before the server starts returning 503s. The default is 1000.
There is an Idle timeout, which defaults to 20 minutes. After 20 minutes of inactivity, the application pool stops. I always turn that off, as it takes time for the first request to process from an inactive application pool.
There is an ability to schedule a recycle at regular intervals or at one or more specific times. Perhaps an untimely recycle is interfering with a long-running operation, and the customer is having requests be canceled as a result.
Those are all of the usual suspects I can think of.