Possible Clues:

• Arachnid of song
• Waterspout climber of song
• Nursery rhyme character who "climbed up the waterspout"
• Determined climber of rhyme
• Nursery rhyme climber
• Waterspout climber
• Determined one in a kid's song
• Kiddie-pleasing song
• Tenacious bug of rhyme
• Tenacious bug?