Lots of great information already posted in this thread but I'll add my bit.
First off, if your first post you mentioned a MiniMaxx as an option. Well it is a great tuning option and platform for custom tuning through MCC it's not an option for you as they don't support 2013+ Cummins. So that can be taken off the table.
As far as tuning goes custom is the way to go no question. I've run custom tunes for almost the entire deleted life my my truck (which has been most of it's life). I switched to box tunes for about 20 minutes and that was all I needed to conclude that they were junk and that custom tuning is the only way to go.
For tuning options the main things you should be looking at are EFI Live or a RaceME which will allow you to custom tune through their new WARP tuning software. EFI Live is one of the top options no doubt. Guys like Ryan Hardway and Kory Willis are doing great work. They were essentially the brains of the H&S tuning before they closed their doors. They know their stuff and they know how to tune a Cummins without a doubt.
RaceME is a newer tuning company in the grand scheme of things but they've hit the market hard and developed a really good product. Their new RaceME Ultra tuner is a really great option for guys that want either box tuning or custom tuning, the ease of a plug & play type tuner without the back and forth, the option for custom tuning, and a touch screen monitor to have some extra gauges visible. Guys like Ray Rich at Double R Diesel are having some great success with custom tuning on the new RaceME's. I personally run custom tuning on my MiniMaxx from Ray at Double R Diesel and am very happy with everything him and I have worked through together.
Transmission tuning is key with whatever setup you go with. The stock tuning on the 68rfe's is junk. They shift way too early to try to keep RPM low but that's exactly what you don't want for the life of the transmission. I have spent far more time working with Double R Diesel on the transmission tuning side of things than the engine. At this point I've pretty much changed all shift points, increased the line pressure, and changed pressure building curves.
At the end of the day how you drive it is key. As mentioned previously gears 1-4 are pretty stout and can take some abuse. It's mashing the throttle in 5th or 6th that will slip the overdrive clutches and fry the transmission. If you drive the truck smart it will handle a fair amount of power. I've been running max performance tunes (around 200 hp over stock) for a handful of months now with no issues.
Now segwaying into the power level. You can certainly run a 20-40 hp over stock tune as a daily driver. Really that works better for a lot of things. When I tow heavy for a longer trip I load a 40 hp over stock file as anything more than that starts building too high of EGT's because of the extra fuel. So I'd say a 40 hp tune is a lot more practical if you just want to tune it and leave it. Now that being said, if you're running a 150-200 hp over stock tune it still drives normally if you're easy on the throttle. You really only notice the difference when you start to lay into the throttle.
The beauty of EFI Live and a CSP5 switch is that you can go from a 40 hp over stock to a 200 hp over stock tune with the turn of a knob. So you can have the best of both worlds very quickly and easily.
Overall tuning and deleting is a world of difference over stock. The 2013+ trucks don't get as much of a fuel economy gain as the 2007.5-2012 Cummins did as the design is different. But the driveability and performance of the truck alone makes it fully worth it. You won't want to go back to a stock truck after driving a deleted one. Getting rid of the torque management is also a great thing as that alone makes the truck feel more powerful. On a 2013+ I would recommend getting the BD throttle sensitivity booster as well as they have a pretty bad dead pedal without it.
