Dangers and threats, moral dilemmas and temptations, duty and responsibility, increased pressure and ticking clocks, miscellaneous challenges

Working undercover to gather evidence against criminals
Needing medication or supplies from a hospital and having to feign an illness to obtain them
Posing as a daycare worker to gain access to children
Joining an opposing sports team to sabotage them
Hacking into a rival company's database to obtain information about their products
Joining an opponent's political campaign to discover their strategies
Gaining employment at a company with the intention of accessing information
Working with a celebrity so the character can rise to fame themselves
The character seeking to gain access to a secret club to increase their social status 
Joining a cult to get a family member out of it

Having to come up with a plan of attack
Needing to build a careful persona before attempting to join the group
Having to study the groups beliefs and rituals to fit in
Others in the group being unfriendly or unwelcoming
Being rejected and having to try again
The character accessing the group easily but not being able to get what they need
The character having to lie about their motives, desires, values, past, etc.
Getting found out and ousted
Breaking character
Needing to change tactics
An insider becoming suspicious about the character
Lacking knowledge of the group's language, culture, practices, etc.
The character having some kind of evidence that could expose them if others discovered it
Not having the skills needed to remain part of the group
The character having to keep their true goal and motives secret from close friends and loved ones
Tension with family members who believe the character is abandoning their values (because the character can't tell them the truth)