Transistor gain, what if there is not enough current?

I understand that a transistor will amplify current by a certain gain factor if it is below saturation.

Good.

But what if there is a resistor that reduces current?

Then the transistor will drive into saturation.

There are many times when this is useful - most notably in the NPN low-side switch. Here we want the transistor to behave like a switch so we inject a base current high enough to ensure that the transistor saturates so that the collector-emitter voltage, V_CE is as low as possible. This eliminates variation in load current due to variations in the transistor gain and also ensures that power dissipation in the transistor is at a minimum.

Figure 1. A typical NPN low-side switch can be driven into saturation by making R1 low enough. Source: LEDnique by the author.

To ensure saturation it is normal to assume a much lower current gain than the h_fe paramater might suggest. 10 to 20 is typical. See the linked article for a worked calculation.

The purpose of biasing a transistor amplifier is to make sure that this problem does not occur. The resistor values are selected so that without any input signal (other than the dc bias) the output voltage will be about half way between ground and Vcc. That setup allows for the largest possible linear amplification.