John
I would think this is ideal for the CA 2A Leveling Amp.

Agreed - And maybe an expander/gate to clean up the low-level stuff.  

Kev999
I wouldn't leave the entire recording on a single track hoping for a one-size-fits-all solution. I would split off the problematic sections (e.g. noisy kids interruptions) onto a separate track and treat them separately.

Interesting thought, but curious - while I have no problem chopping up a guitar part so that it only plays when the guitar is playing, that is in the context of a mix.  If I was recording say an acoustic guitar, I would expect string noise, other hand noise... to be part of the recording.  As such, how do you do this with spoken dialog? 
 
If spoken dialog is coming from -100 db to 0 db (or whatever) by chopping out the unimportant parts (kids...), wouldn't that come across somewhat unnatural?  Would you recommend a low level tape hiss or the likes to provide a more realistic reproduction?  
 
I still think a leveling compressor is needed to tame the extreme peaks.  

ardjunc
Other than cleaning up the occasional pops and clicks as the recorder was being turned off and on and perhaps some overall eq I am not convinced there is much to be done with the quality of the recording. Picture a cheap mic sitting on a table with several people talking then the kids come in and start competing for attention. The recording is fairly listenable as is, my goal is to level things out which is why I gravitated to compression first then apply any enhancements later.

I'll experiment with different types of compressors and settings. I have always recorded the best possible take in the first place so not much had to be done in the mix. This project will certainly provide invaluable experience in how to "fix it in the mix".

It never hurts to tackle the worst to learn the quickest.

Ardjunc

There were plenty of memorable recordings done with not much more.  The recordings of Robert Johnson are terrible by today's' standards, but inspired generations of blues and rock players.  

space_cowboy

Kev999
I wouldn't leave the entire recording on a single track hoping for a one-size-fits-all solution. I would split off the problematic sections (e.g. noisy kids interruptions) onto a separate track and treat them separately.

Interesting thought, but curious - while I have no problem chopping up a guitar part so that it only plays when the guitar is playing, that is in the context of a mix.  If I was recording say an acoustic guitar, I would expect string noise, other hand noise... to be part of the recording.  As such, how do you do this with spoken dialog? 
 
If spoken dialog is coming from -100 db to 0 db (or whatever) by chopping out the unimportant parts (kids...), wouldn't that come across somewhat unnatural?  Would you recommend a low level tape hiss or the likes to provide a more realistic reproduction?

 
Whatever compression is used on the main sections isn't going to work well on the kids sections. That's why I recommend treating them separately. I'm not sure about whether it's going to sound "natural".