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.

• Wing-Nut 452 Wire Connector, Red. Which accepts 1 to 6 #14 conductors.
• Twister® PRO Wire Connector. Can also handle 1 to 6 #14 conductors.
• Wing-Nut 454 Wire Connector, Blue. Can take 5 or 6 #14 conductors.
• Twister 342 Wire Connector, Gray. Allows 3 to 6 #14 conductors.

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:

1. Remove a bit of the insulation in the middle of the wire.
2. Wrap the exposed wire around the terminal screw of the first device, and tighten the screw.
3. Remove a bit more insulation further down the wire.
4. Wrap the exposed wire around the terminal screw of the next device, and tighten the screw.
5. 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.

Daisy chain

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.

As long as all the equipment is listed for the use, there's no problem using stranded wire. The UL White Book says that screw terminals and pressure plate terminals can be used with both solid and stranded wire.

UL White Book 2013

Receptacles(RTDV)

Receptacles for Plugs and Attachment Plugs (RTRT)

Terminals
Terminals of the wire-binding screw, setscrew, or screw-actuated backwired clamping types are suitable for use with both solid and stranded building wires.

It says switches are solid only, however, you can use terminal connectors such as fork connectors.

Switches (WFXV)

Snap Switches (WJQR)

Terminals
Terminals of the wire-binding screw, setscrew, or screw-actuated backwired clamping types are suitable for use with solid building wires unless otherwise indicated either on the device or in the installation instructions.

Terminals of a flush snap switch are permitted for use with Listed field-installed crimped-on wire connectors or an assembly, if so identified by the manufacturer.

A flush snap switch may also be provided with conductor leads with factory-installed crimped-on connectors. Such connectors may be either attached to the flush snap switch terminal or are provided with the flush snap switch in the smallest unit shipping container and are suitable for use with the terminal of the flush snap switch.

Wire connectors (including twist-on wire connectors) can be used for both solid and stranded wire sizes between 30-10 AWG, in combinations listed by the manufacturer. Unless they are clearly marked to be used with one type or the other.

Wire Connectors (ZMKQ)

Wire Connectors and Soldering Lugs (ZMVV)

Wire stranding -- Unless clearly marked "Solid," "SOL," "Stranded" or "STR" for a given wire size, wire range or wire combination, conductors in the range 30-10 AWG are both solid and stranded, and 8 AWG and larger are for stranded wire only.

Ideal Industries has a handy PDF, that lists all the allowable wire combinations that can be used with their various connectors. For example, the Ideal 74B® (their basic yellow wire-nut) can be used with 1 - 4 #14 solid or stranded conductors.