Hotel chains put up a lot of signs telling you how "eco-friendly" they are, but really what they're doing is cutting energy & labor expenses from the budget while reducing your comfort level and ability to control your environment.
The systems they commonly put in place to limit the air unit are example #1. Fortunately, these are also usually trivial to hack. They tend to have a simple, standard way to get into "maintenance mode" and from there it's just a matter of knowing what the codes mean.
Thus, this guide to the ones I've most commonly encountered over recent years of travel (at least in the US).
Of course, if you just want the white noise of a constant fan you might consider investing in a good portable white noise machine instead.
Amana
Amana's under-window units seem to be the most common ones I encounter in hotels everywhere I go. They have some variety to their setup, though. The base unit is always pretty much the same, but some are controlled from a wall thermostat while others have the controls right on the unit (the latter is much more common in my experience; the wall thermostat is an upgrade module the hotel has to install that costs extra money).
These things often let you just run the fan endlessly by turning the unit off, then pressing the "fan speed" button to your desired setting. However, I've encountered at least a few properties that don't. Also, they often have temperature setting restrictions that are too narrow for extreme summer or winter conditions.
If the unit is controlled directly by the panel on the unit itself, you usually go into maintenance mode just by holding down the + and - temperature adjustment buttons together and then tapping Off either once or twice. It's in maintenance mode when you get a null value (two -- lines) on the display, usually with a blinking dot beneath.
From here you press the Heat and Cool buttons to cycle through the various settings, represented by a C(number). C2 is the setting for the constant fan. Wait a second and you should see it toggle to "Au", which means it's in automatic shutoff mode. Press the + key to cycle it to a capital A by itself (not bA). That's the "always on" setting. Then just press Off and you're sweet.
If you want to see what other settings are available, check out the service manual.
Verdant / Friedrich Wall Units
These are units that are usually mounted under a window, but have a thermostat in the wall elsewhere in the room with a digital panel that you control them from.
The digital panel should have a small screw at the top that holds the plastic face on. If you don't have a screwdriver, a nail file will work (and can be carried with you if you're taking planes). At some hotels you'll find the screw has already been removed as it's not necessary to hold the plate in place, and maintenance frequently accesses these things.
Pull gently from the top to remove the front plastic cover. In the upper-left corner you'll see two tiny holes. Use a unfolded paper clip, toothpick, pin, jewelers screwdriver or something like that to poke into the uppermost one. There is a tiny button in there you have to keep held in to alter the settings. While held in you should see the normal temperature display change to a "01" code.
If you want the fan to run continuously, press down (to 00) then the power button. You can also flip through some other settings with the F/C button - heat limit, cooling limit, and the "night occupancy" settings being the most interesting ones. See the manual for further details.
The case should snap right back into place easily.
Honeywell / Inncom
These are wall-mounted units that you'll see commonly in hotels that have vents built into the walls, somewhere in the ceiling or in a column in one corner of the room. You see these at a lot of casino-hotels in Las Vegas and other places (Caesars seems to use it at most of their properties), and they're also common in Marriott locations.
With most models, you hold down the F/C selector button and press OFF once, then DISPLAY once. With some older models, you simply hold down DISPLAY and press OFF instead. When you let go of the F/C (or DISPLAY for the older models) button the temperature display should backlight and change to a 0.
It will then flicker back to the normal temperature display, but you now have greater control over the temperature and fan. Turning the fan on will keep it on indefinitely, and the temperature range will expand in both directions.
If you're doing a long stay, just be aware that this mode resets back to normal with some units at some point, usually 48 to 72 hours after being activated.
GE
This unit is very common at Holiday Inn properties, and I've seen it at some Comfort Inns too. This is usually an Amana-like wall unit with a separate wall thermostat control somewhere else in the room; the control panel on the unit is usually blocked by a sign telling you to go use the wall control.
With this one, you actually have to pull the whole front panel off the wall unit. Don't worry, this shouldn't require tools and should come right off and go right on easily. You push the Aux Setting button under there, then lift the sign blocking the unit control panel (which should now be active) and press Mode to get to the desired setting. There should be a list of settings posted right there on the unit, but if not it's four presses of Mode to enable constant fan and five presses of Mode to get to the temperature limits.
More Air Units?
Of course that's not every unit out there, but that covers some of the most common ones that I keep seeing. If you want to send suggestions for units to add, please do, I'm happy to update this article over time.