Crothix, Ritathix DOE, and glycol distearate.
I encourage you to read the links in this post about increasing mildness and viscosity for more details on this topic.
You definitely want to include a thickener in your product as pretty much everything I make from surfactants on this site is about as thick as water before adding one. I prefer to use Crothix in my products, and it offers a thickening as well as a moisturizing feature.
To be honest, I'm not a fan of Ritathix DOE. It felt sticky on my skin. You can also use regular old salt to thicken, but's a picky process and some surfactants will not thicken this way. Check before using.
So do we need thickeners? A resounding yes! They thicken the product, offer moisturizing, and increase mildness.
If you'd like to see thickening in action, check out this video on my YouTube channel!
Other posts in this series:
Shampoo - How does it work?
Shampoo - What's in it? Surfactants
Shampoo - What's in it? Other ingredients
Shampoo - Increasing mildness & viscosity
Shampoo - Conditioning agents
Shampoo - Dimethicone
Shampoo - Proteins and amino acids