Calculating mean of data frame inside a series object
In the absence of a MultiIndex (the Right Way$^\mathrm{TM}$), the apply method can do what you want; e.g.

df.assign(
    max_retweet=df.tweets.apply(lambda x: x.retweet_count.argmax('retweet_count')),
    avg_retweet=df.tweets.apply(lambda x: x.retweet_count.mean())
)


       avg_retweet  max_retweet  
26662     0.045476          187  
32316     0.821538          427  
25879     0.633681          583  
43411     0.112465          421  
28840     0.472222            8


Also, don't share data with pickle; it's dangerous.