MSP430 above 3.6V

If you're confident that the battery voltage remains within the MSP430's specs you don't really need the regulator, but I would recommend to use one to have a clean power supply voltage. In any case, be sure to have a look at other components the MSP430 connects to. For instance when you drive a MOSFET the gate voltage may be below pinch-off and the FET won't do its job.
(I don't quite remember the specs, but I once tested how low the MSP430 would go, and found that it still worked at 1.3V. Don't take this value as a reference, though. Always refer to the datasheet!)

If you want to measure the battery's voltage you'll need an ADC reference voltage that's independent of that. For instance use an LDO to create a fixed ADC reference voltage of 2V, and divide the battery voltage by two before applying it to the ADC (to make it lower than the reference)

The \$V_{CC} + 0.3V\$ is Absolute Maximum Ratings. It's an often made error, but you're never supposed to operate your device under these conditions. Even if the numbers suggest that these are acceptable values, continuous operation at those values may damage the device.
That said, most types of ADC, including MSP430 sigma-delta, will give a maximum reading if the input voltage is higher than the reference. The reason is that the ADC will compare a voltage derived from the reference (either through integration, like in sigma-delta, or through charge redistribution, like in SA) against the input voltage, and since this will be higher than the reference this voltage will never reach the input's level.

edit
It's good design practice to have a keen eye for such maximums. If you know that the input voltage may go higher than the ADC reference, you better scale it down a bit using a voltage divider. Not only do you prevent damage, but you'll also be sure to be able to measure over the full range. If the reference voltage is 3.6V, then an input voltage of 3.6V and one of 3.7V would give you the same reading, but you won't be any wiser.