Thank you very much so far for your long answer.
I'm glad you try to help me!
1) Never a good idea to short circuit a transistor, regardless of rating.
Why not?
2) I doubt you will find this exact configuration you have in ANY amplifier designs
Yes, I know, I only build this to measure the current.
3) Never assume there is some "inner resistance" in a transistor.
How else would you tell it?
In short, I feel that you have less understanding about NPN transistors than you believe you do.
That's right I don't know much about any component. Very new to it.
Are you sure you mean "amplify"?
As you say it... No? I'm sorry, my English has got rusty. I don't want to amplify something like an audio signal. I just want more power than the arduino can provide. Like you say, an ON/OFF SWITCH . I'm using a timer interrupt to switch the LED ON/OFF every 13 us to achieve the 38 kHz the reciever needs. It already works on my breadboards but not with the desired current.
How about we give an example of how you can get an NPN to amplify? Good place to start? Maybe I will draw something and do a follow up post.
It would be great if you could help me to solve my problem.

A problem you have right now is this. A transistor can SWITCH ON and OFF and it can amplify... sometimes it can both... but it's related to base biasing. Using just 1 resistor won't really do it for you. You *can* make a voltage amplifying switch this way, (if the collector V+ voltage is higher) though there are some issues with how you have things connected. in other words... you have sort of the right idea... but a less than optimal hookup.
Yeah unfortunatley I don't get what I have to change yet. Do you understand me at least? I want this IR LED to be pulsed at 38 kHz. In the datasheet there is a description that says that you can use it with 3 A (for 10 us in the example). Because you don't get that much from an arduino pin I wanted to use a transistor. But I only get about 100 mA.
Ok, rather than draw up something new... I'll give you a drawing I already created that does what you want. Obvuously, change out my values of transistor and LED for yours.
At first view I would say I've done it this way.
So why do I get only 100 mA when I short circuit a transistor? How do I get the transistor to let the whole current pass?
Do I have to use a MOSFET instead?