Why my hex inverter chips (all three of them) are not producing a HIGH output with low input?
If you look at the data sheet for the 74LS06, you will see the following.

Note it says the outputs are open-collector. That means it can sink current but can not source it. (Well it can actually source a little current as leakage or more if you pull the output below the bottom rail.)
Looking at the internals confirms this.

If you compare that to a push-pull output like the 74xx04..

You can see that the output has two transistors. The bottom one is similar to the output of the open collector device and is turned on when the output is low and pulls the output down, sinking current. The top transistor is turned on when the output is high, it sources current and pushes the output high. Hence the name push-pull output.
On an open-collector device, with no pull-up on the outputs, you will not be able to measure much voltage on the outputs when the outputs are high.
You can use the open-collector device to pull current down from Vcc through an LED if you so desire, but do not exceed the maximum output low sink current.



simulate this circuit &ndash; Schematic created using CircuitLab
As Gee Bee mentioned, you can also do it, without the inversion, by shunting out the LED, as shown below. However, this method wastes power since current will always be flowing through the resistor. It actually uses more power with the LED off than when it is on. So I do not recommend this method.



simulate this circuit
Do not try the shunt method with a push-pull invertor though.