Title says it all...

I would like to plug my device to the charger and let it be running the entire time (just like a laptop) but without battery.

Is it possible? If not (I guess so) why?
What can I do to make it possible?

Nope, while in use the N900 often uses the battery quicker than it charges.

There is a thread on here about 'changing the battery without turning it off' but its not a good idea.

the thread
http://talk.maemo.org/showthread.php?t=57734

in theory will work.
charger asic will keep the voltage across battery terminals under 4.2 volts and n900 current consumption will keep charger chip active .
the only reason for not working in this mode is bme software
max voltage of twl4030 chip I think it is 4.5 V ( according tps65950 datasheet ) so if bq24150 keep the voltage under this value it is ok for running.
anybody want to try 4V high power zenner diode ?

the rate the battery of the n900 charges doesn't matter. the rate of charging isn't because the outlet can't deliver more, it's because the battery can't charge quick enough. He want's to do away with the battery altogether so this limiting rate is gone. Probably an external power supply is enough. The question is why do you want this?

charge battery and keep charged by running without it

Leave it connected to my TV and network.
I could put some BT keyboard (and possibly some mouse), rygel and transmission. Then use it as a media player, browser and mail for the tv.

For that I would like to know if it is possible to run the device without battery (since at some point it will die and I do not want to spend with that if its possible to leave it running all the time).

I think it's probably far better just to leave the battery in your phone. You'll have to replace it after about a year or two whether you use it or not anyway. They all fail in some way eventually. Always.

I'm not sure if things are the same where you live, but in Asia you can buy an original brand new BL-5J for about $15 USD. Easier just to replace it. If you want an identical battery from a 3rd party, then you could get one of those for about $3 to $5 USD.

www.youtube.com/watch?v=sa-WNmsLMaE

as we see it can be do it....but i also see that nobody in maemo.org dosnt want to explain it, no ?
its nice to use without battery, why not ? i see guys telling about to buy a new battery or to change hotswap - but withot battery, using cable as laptop, its not good? if there have ideas please, we'r all wait