Voicemail app that doesn't require carrier voicemail I'm looking for a voicemail app that will answer my phone after a configurable number of rings and route the caller to voicemail stored on the Android device. 

I don't have voicemail with my carrier.

I'm not interested in features of Google Voice because it seems to piggyback on real voicemail and also I don't want to port my phone number to another service if indeed that is required. 

(Note: I remember doing something similar on a PC desktop years ago when the phone line was hooked into the modem, the computer would act as voicemail.)
Try if any of these apps work for you. It would depend on your phone, as android has limited support for total offline answering machine capabilities due to technical barriers. See this thread on XDA. If your phone isn't compatible, you might have the following issues:


Not be able to record calls
Record the conversation at only one end
Very low sound quality or volume


Google voice doesn't depend on your voicemail. The calls are recorded on google voice's servers.