The stock alternator is truly undersized for the application as it puts out only 65 amps at hot idle and 100 amps when driving down the road at full operating temperature. It's the same part number used in the Ford Fusion! Further complicating things is the 230 amp draw caused by the glow plugs primarily right after the truck starts - check out this Fluke log compliments of user MisterCMK on FTE. So...if your alternator is putting out less than 13.3 volts at hot idle (as measured at one of the batteries) with all accessories (headlights, blower motor, heated seats, inverters, radio, etc) on, consider moving to a Leece Neville or DC Power alternator.
The 185 amp OEM series puts out 100 amps at hot idle and is a solid choice for you if you don't run a plow, winch, or serious inverter and live in a southern climate. If you live in a colder climate, consider how much of the 230 amp initial draw from the glow plugs, etc running at startup you want to cover (the more, the merrier), but minimally look at the 190HD. If you want to bump up higher given that most F series trucks have about 100 amps of gear that can be turned on from the factory and you are concerned about ever drawing down the batteries just to keep the engine running, move up even more - perhaps the Leece Neville 230 amp or even the DC Power 270XP or 370XP..
If you DO run a plow, winch, or large inverter, the Leece Neville 230 or 270XP are the alternators of choice. They are more money, sure, but you can run small cities off of the 170 or 180 amp, respectively, they put out at hot idle. Wow.
Of course, you should also consider coupling any alternator with auxiliary alternator wiring (see point 3 on this list) since the stock wiring size results in an excessive drop in voltage by the time it hits the batteries. A great discussion on all this, including the source of the above Fluke log can be found on FTE. DC Power posted a video showing the output of the 270XP unit as well.
Don't worry about over-charging your batteries with any of these options - the internal voltage regulator will only give the system the oomph it needs up to the capacity of the alternator but never more.
If your alternator is putting out the right voltage, but you are still concerned and just want a reason to move into a new alternator...we can help there too. We've seen many a FICM's logic board get completely fried by a bad voltage regulator - a common failure in any alternator even IF the voltage output measures within spec during your voltage test. It's a sad little reality.
Most 6.0 owners will tell you that the sweet spot for failure in a 6.0's factory alternator is between 120,000 and 160,000 miles.