You can minify most of your Arduino projects using the ATtiny85, if you know how to do it properly. Here is a step-by-step guide to programming the ATtiny85 using the Arduino Uno as programmer.
Components List
Here are the components that will be used in this experiment:
- Arduino Uno
- ATtiny85 Chip
- 10uF capacitor (10uF to 47uF can be used, actually)
- Breadboard and Jumper Wires
Step 1: Set the Arduino as an ISP Programmer
Go to File > Examples > 11. ArduinoISP > ArduinoISP.
Click the Upload button on the Arduino IDE and wait until the program is successfully uploaded. After this, disconnect the board from the computer.
Step 2: Add ATtiny Boards to the Board Manager
The ATtiny series of boards are not native to the Arduino IDE so it must be added through the board manager. To introduce the ATtiny boards to the Arduino IDE, you have to go to File > Preferences.

On the Settings tab of Preferences, click the icon on the right side of Additional board manager urls and paste the link provided below:
https://raw.githubusercontent.com/damellis/attiny/ide-1.6.x-boards-manager/package_damellis_attiny_index.json

After adding the link click OK and the open the Boards Manager of the IDE.
On the Search Bar, type attiny. Look for attiny by David A. Mellis and click install.

After installation is done, you may proceed to wiring the ATtiny85 with the Arduino UNO board.
Step 3: Connect the ATtiny85 to the Arduino Board

Connect the ATtiny85 to the Arduino Uno board as shown above. Take note that you may use either a polarized or non-polarized capacitor with a value from 10uF to 47uF. Be sure to follow the correct polarity if you are using an electrolytic capacitor: anode must be connected to RESET and cathode must be connected to GND.
Step 4: Burn the bootloader to the ATtiny85 chip
Do the following in the Tools Tab of the Arduino IDE:
Board > attiny > ATtiny25/45/85
Port > COMX(Arduino UNO)
Clock > Internal 8 MHz
Programmer > Arduino as ISP
After doing all of the above, click Burn Bootloader
Once burning Bootloader is done successfully, you can proceed to uploading your sketch to the ATtiny85 chip.
Step 5: Upload your code
This is where a lot of people get stuck. Uploading the sketch cannot be done by simply clicking the Upload button on the IDE. Instead you must go to Sketch > Upload Using Programmer and wait until the program is successfully uploaded.
Step 6: Test your circuit
Once the upload is successful, you may remove the connections to the Arduino UNO board and just simply apply power to the ATtiny85 through VCC at pin 8 and GND at pin 4.
Here is a video of Blink running of the ATtiny85 using the Arduino Uno only as a power supply.
With ATtiny85 and a bit of creativity, you can build a lot of interesting circuits. Tell us what you want us to build next.