I am particularly interested in copying several chunks of text and then paste all of them together into another app
I believe what you are looking for might be a Clipboard manager. There are a few around for OS X (e.g. https://itunes.apple.com/au/app/copyclip-clipboard-history/id595191960?mt=12) - you can easily find more on Google.
Now, for iOS I don't believe you can have that exact same functionality, but some apps get close. They can check your clipboard content and you can add any copied text into that app easily as shown here: http://www.cleanshavenapps.com/clips/
Another popular app is found here: https://itunes.apple.com/au/app/clipboard-manager-history/id320175134?mt=8
I don't think any of these will merge your copied texts automatically, though - you could check out IFTTT to see if you can create your own (or existing) automated recipe. Maybe also Workflow or Launch Centre Pro could help automate this for you.
Thanks Chris for the suggestions. I found another app called NoteBox, recommended by Federico Vittici but not tried by me, that appears to do something like it.
I also found a small (and old) one trick pony app called Copy+ HD, last updated 2012 that seemed to have been designed exactly for what I wanted, so I splurged and bought it, even though I feared it would not work any more. Well, it does! It's not perfect but it is closer to this task than any other app I know of. (I did not try Workflow or IFTTT.)
Paste is a contemporary alternative with an iCloud synced clipboard. It's possible to concatenate multiple items from the clipboard history. In addition to recent clipboard history, it offers "Pinboards" where important clips can be saved, e.g. code snippets, folder paths, images & documnets and so forth. It's possible to preview the files from within the app.
There is a trial version available.