Use the right connector
Ideal Industries makes a few different varieties of twist-on wire connectors that can handle 6 conductors. After looking through the UL Listed Combinations (PDF) document, I was able to find 4 such connectors.
If you find yourself needing to connect more than 6 conductors, it's probably time to start looking for something other than a twist-on wire connector. As @Aaron pointed out, you can use a Push-In type connector for up to 8 conductors. You might also want to consider using a Crimp Connector, which can connect 4 to 10 #14 solid conductors (an insulated cover would be required when joining current carrying conductors).
Moving beyond crimp connections, you'll likely have to start looking at terminal blocks, bus bars, design changes, or alternative solutions.
All conductor combinations will vary from manufacturer to manufacturer, check the manufacturers documentation before using any electrical connector
Use devices as connectors
Another option, is to use unused terminals on devices as splice points. For example, you might have a switch with both side and back terminals. Using the back terminal as a splice point, is an acceptable way to extend the circuit. In fact, some industrial grade receptacles (like the Leviton 5252 Series) offer 8 clamp style terminals on the back of the device.
Reduce the number of pigtails
Where multiple devices share a single hot wire, you can reduce the number of wires in a twist-on type connector by using a single extra long pigtail. You'll use the extra long pigtail to connect all the devices, eliminating the need for a single pigtail per device. So you can take the number of required pigtails from 3, down to 1. If you leave the feed hot extra long when originally wiring the circuit, you may be able to eliminate pigtails altogether. To do this:
- Remove a bit of the insulation in the middle of the wire.
- Wrap the exposed wire around the terminal screw of the first device, and tighten the screw.
- Remove a bit more insulation further down the wire.
- Wrap the exposed wire around the terminal screw of the next device, and tighten the screw.
- Repeat until all devices are connected.
Once you have all the devices connected in this manner, you can use the end of the wire to feed through to other devices. Simply remove a bit of insulation at the end of the wire, and use a twist-on wire connector to connect this wire to the wires feeding the other devices.
As for connecting bundles of wires with pigtails. There's no problem doing this, as long as you don't exceed individual connector conductor fill, or overcrowd the box.