This is a recurring question and will never be answered to everyone's satisfaction. Some things to consider...
1. The 15A circuit in your home already has the safety margin built in to it. You don't need to extend it.
2. The breaker tripping is a nuisance, not a safety issue.
3. LEDs do reduce the current draw of the games by a lot.
4. A 15 amp breaker does not trip at 15 amps. It takes over 30 seconds to trip at 30 amps, so you don't have to worry about short surges. Web search for "15 amp breaker trip curve" if you want details. Fair warning: it takes some time to understand the chart.
5. The raw current capacity of a 14 gauge wire is way more than 15 amps. Electric code limits the voltage drop to 5%. This allows 160 ft of 14 gauge wire in a 15 amp circuit, but code is probably more conservative (see #1).
6. The distance from your outlet to the electric panel will be a factor (see #5).
I've never heard of anyone with 6 pins on a 15amp circuit that blew a breaker but I can't declare it impossible.
- Kerry*
*Electrical Engineer but not a licensed electrician