Looking quickly at Article 408 in the 2008 NEC, the limit of 42 overcurrent devices in 408.36 Exc.2 probably applies as an absolute maximum. 408.54 further limits the maximum to that for which the panelboard was designed, rated, and listed. Counting the devices, a DOUBLE-pole breaker is counted as two devices. The NEC, as best I can determine, is (strangely) silent on the count for TANDEM breakers. I count them as two, but others may differ. In any event the panel maker should have posted the maximum number, if any, of tandem breakers permitted.

According to your panel # you may not use any tandems. A 3030 means 30 spaces, 30 circuits. If it were a 3040 you would have 30 spaces but allowed to have 40 circuits, and 5 of the 15 stabs would be notched, allowing you to use 10 tandems. The notched stabs would probably be the bottom 5. You will also see a label with a diagram showing where the tandems are allowed.

Yes you can buy non circuit limiting tandems which will fit in panels with no notches, but they are intended only for replacements on older panels before the rejection methods were designed, and where you don't exceed the number of circuits allowed by the manufacturer.

As for the NEC, it falls under the requirement to follow manufacturer instructions.

"The NEC, as best I can determine, is (strangely) silent on the count for TANDEM breakers."

The NEC limit is on 'overcurrent devices'.
A tandem / slim / half size breaker contains two devices.
A 240 V breaker contains two devices.
A 3 phase breaker contains 3 devices.

The limit on a residential panel is 42 devices, not including main devices if present.

How would a singular slim or half size breaker count as 2 devices?

How would a singular slim or half size breaker count as 2 devices?

Actually, it doesen't. I think GE is the only skinny made anymore, everybody else does tandem or piggyback types which are two breakers. Brick just kept going on a list of stuff and should have stopped sooner.

There are so many brand names and 'trade' names for the breakers it is ridiculous.

If it has two handles (attached or not) and controls two hot lines it counts as two devices.

If it is a single handle but has a smaller width it is a single over current device device.