I have at least one alarm set to go off daily so the "alarm clock" icon on my notification bar is really pointless.
Is there a way to make this icon hidden in Android Froyo or Android Gingerbread?
No, for the same reasons detailed in this answer.
On Android Marshmallow, you can use the hidden System UI Tuner.
Open the quick-setting panel, then press and hold the Settings' cog wheel until it starts spinning. You then get an additional settings entry in System > System UI Tuner > Status bar, where you can hide the alarm.
Easiest way if rooted: You can use GravityBox, to hide the Alarm-Clock-Symbol.
There is if you use the alarm app, Gentle Alarm. Besides the plethora of amazing features that app has, you can also turn off the notification bar icon and the app still runs in the background just fine. I've been running it that way for a long time.
On a semi-related note, my favorite feature of Gentle Alarm is the ability to keep alarms scheduled, but skip as many of them in order as you want. That way if you are on vacation for one day you can turn off scheduled alarm(s) for just that day and not be annoyingly woken up. I was on vacation for a week and was able to turn them all off and the following week I didn't have to remember to turn my alarms back on, already taken care of.
In Gentle Alarm, you must go to the Settings and turn on Advanced Features. Then Other Preferences > Notifications, and then you are presented with many options for the type of notification or turning it off.
I achieve this with the app; Smart Alarm Free. It simply has a setting for it. Also it's a way better alarm app.
Yes there is, but you need an app for it.
I was surprised to read that this was something people wanted, and so I ended up writing such an app (disclaimer: I wrote the app, but it is free).
It is small and lightweight, and removes the icon, but will also flash the icon every time another app updates the alarm.
If this is something that you would find useful, you can get it for free on the Google Play store:
Just found this in Marshmallow for anyone coming across this question. Go to Settings > Status Bar > Status Bar Icons and from there you can uncheck the alarm icon.
Provided your Android is rooted and a compatible Xposed Framework1 is installed, you can use GravityBox xposed module to hide the alarm icon.
I tested with GravityBox [N] and in it, go to Statusbar tweaks → Clock Settings → tick Hide Alarm icon.
Note that I cannot assure whether earlier Android version specific versions of GravityBox offered this tweak.
(Click image to enlarge)
For Oreo, I used SystemUI Tuner. Need to enable the developer options temporarily and grant it necessary rights though a computer. It exposes the necessary system settings and has few other useful settings too.
Bit more detail: In earlier versions it was possible to invoke the "System UI Tuner" on android system by long pressing settings cog wheel after swiping down on the notification bar to expand it. Sadly it has been decided not to expose this to users on Oreo. So we need a 3rd party app (e.g. first link above). And give that app necessary permissions using developer mode. The two links give all the details and steps.
You can download Wanam Xposed from Xposed repository here.
When you open the app you'll be instructed to install the framework (I've done it personally and haven't had any problems).
Once this is done go back to the original app and there will be many customizable features available including one for hiding the alarm clock icon.
This is not my app. I just use it personally on a stock JB 4.3 (rooted) and find it great.
if you are running Jelly Bean (or newer), there is the option of disabling all notifications from an app. The setting is per-app and can be found in Settings > Apps > (select the app) > uncheck the "Show notifications" box. The one caveat to this method is that the app will no longer show any notifications in the notification bar, but if that's acceptable to you than I've found this method to be an easy way to stop my notification drawer from filling up with unnecessary notifications.
Also, if you disable notifications entirely for an alarm clock, nothing will pop up when the alarm goes off. To turn off a ringing alarm, you will have to start your alarm clock app from the home screen, find the alarm schedule that is currently ringing, and turn off the scheduled alarm. You will have no option to snooze the alarm, or silence it but leave it on repeat. You'd have to turn it off and then on again.