Add the cubed potatoes, diced onion, minced garlic, and grated ginger to the slow cooker.
Stir in the red curry paste, coconut milk, vegetable broth, salt, and black pepper until the curry paste is mostly dissolved.
Cover and cook on low for 6 to 7 hours, until the potatoes are very soft.
Use an immersion blender to puree the soup directly in the slow cooker until smooth, or transfer in batches to a countertop blender.
Taste and adjust salt or pepper if needed.
Ladle into bowls and top with chopped cilantro. Serve lime wedges on the side for squeezing.