Degree + Connections + Career Hopper Bomb

No, the highest you can start a new career is level 8, with a Distinguished honors degree.

This is how things work for the base game, based on my experience. None of these stack:

• Sims who had an A in high school when they grew up to Young Adult start all careers at level 3. This includes restarting a career that they've already had, if they didn't choose a branch yet.
  
• Sims with the Connections reward start a new career at level 4. This doesn't work for restarting careers they've already had.
  
• Sims who have already chosen a career branch will restart a career at the beginning of that branch.
  

Get Famous

• Sims with the Career Hopper reward start careers at a higher level than normal. I don't know how much and I don't know if it stacks with any of the bonuses from the base game.
  
• Sims with the Career Hopper reward restart careers they've already had at the same level they had before.
  
• Sims with Good, Great, or Pristine reputations get frequent job offers over the phone, with starting levels around 4 or 5.
  

Discover University

• A normal degree lets you start the career branches it applies to at level 6, or at the first level of the branch career if that's higher than 6.
  
• A normal degree with honors lets you start the career branches it applies to at level 7, or the first level of the branch career if that's higher than 7.
  
• A distinguished degree lets you start the career branches it applies to one level higher than a normal degree but no higher than level 8.
  
• A Distinguished degree with Honors lets you start the career branches it applies to at level 8.
  

It's been reported that University degrees override the Career Hopper reward, so you might not want one if you've already reached the top of a career.