The tech can't add an extra custom field on the fly.
It is possible to make it so they could add an Approval Process, but it would take some work and could potentially get to be exponentially more of a pain to set up with the more variables you have.
The way that would work is to create a custom field like NEEDS APPROVAL PROCESS or something like that and make that custom field hidden from Clients but Editable for Techs. You'd make that field a "Pop Up Menu" field with options something like this:
No approval needed
Approval Process 1
Approval Process 2
You would assign that Ticket Custom Field to all of the tickets which might potentially neet to get an approval processes assigned to them.
Then you would configure a unique Action Rule for each one of those options in the pop-up menu that has settings like this:
[Action Rule Info tab] Evaluate Criteria... only when updated by a Tech
[Action Rule Info tab] When Criteria evaluation results in a match, apply action... only if the criteria did not match before the Ticket was udpated
[Criteria tab] Custom Field NEEDS APPROVAL PROCESS Is equal to Approval Process 1
[Actions tab] Add Approval Process Approval Process 1
Of course you'd also have to define the approval processes as well.
You can make a task, make the task element run an approval process, and then create the approval process and tie it to the task. Then you run the task on the ticket from the Group Tickets view. Or, you could do Kelly's idea from above, which doesn't require a separate process but requires an action rule tied to a custom field instead. Depends on whether you want to change your custom field display order or not as far as I can tell.