Good question! Here some ideas:
- Solving complex problems and structuring uncertainty
- Working with a highly capable, small team and learn to achieve high output in short time
- Understand typical problems large corporate clients face in today's environment; get involved in high-stakes topcis/projects/decisions (as opposed to joing large companies in line functions)
- Getting to know how decisions are made in big companies and how they work
- Build technical skills: Building models, developing and structuring top-management presentations, presentation skills
- Learn fast by continuously iterating through new projects in new teams, and getting instant feedback from team members
While the question is clearly about what you want to learn, I think it's never bad to also think and add how you can contribute to the consulting firm's and clients' success while you are learning.
In the end, consulting is no business school and clients pay the bill ;)
All the best!