This category differs from the others somewhat. Urgent situations are rarely planned, and sometimes involve people who aren't already close. This can make them good for introducing characters to each other. Conversely if they happen between people who do know each other, they tend to change the nature of the relationship. Also, such urgent situations happen infrequently in everyday life, but they appear more frequently in the high-tension atmosphere of fiction.
Childbirth. Attending the blessed event entails providing a lot of moral support for hours under high stress. It can create a bond with the baby as well as with the mother. When planned, this opportunity is only offered to the closest family members or friends, barring professionals. But it can happen by surprise in very awkward circumstances, a popular motif in fiction.
Saving someone's life. Quick action in a life-threatening situation demonstrates how much one person values another. This can create a strong sense of connection, and sometimes obligation. It often, though not always, entails personal risk for the rescuer. This is fairly typical for military buddies or police partners, etc.
Risking your life for someone. Placing someone else ahead of your own life shows their importance to you unequivocally. This often, though not always, involves trying to save or protect another person. While it can create a sense of gratitude, it frequently causes anger as well -- someone who loves you will generally object to you endangering yourself, even to protect them. Military and police buddies protect each other regularly.
Making emergency decisions for someone. This reveals both how well you know the person, and how much you care about them -- whether you know what they would want, and act on it even if it differs from your personal preference. Unlike some of the other options, in this one the initial action is often outweighed by the aftermath. Both characters have to deal with the results of the decisions, good or bad.
Deathwatch. Dying can be as intimate as giving birth. Staying with someone while they pass is an act of love; so is providing moral support to someone sitting deathwatch for a family member or other person. Many soldiers and police have done this for someone.