Characters have a need to feel competency. Once all of the other basic needs have been met - enough to eat and drink, a sense of security, and acceptance from peers - it is natural for human beings to seek out opportunities to feel good about themselves. To feel important and clever. In literature, this type of conflict is difficult to describe because it tapers away to questions of love and acceptance on the low end and issues of ethics and morality on the upper end. Also, I think many people have a mixed reaction to obviously talented people having those abilities tested. On one hand, professional sports wouldn't exist if we didn't, on some level, enjoy watching talented performers push themselves in competition. But on the other hand, we tend to mock or revile those people who seem too proud of their own accomplishments or too flashy in victory. "Don't spike the football."
This is a tough balancing act for Star Trek: The Next Generation. Put simply, every single character comes off as incredibly arrogant at some point in the first season. Whether it's Picard's deliberately unironic reading of Hamlet's "Piece of Work" monologue, Riker's high-handed treatment in "Angel One" of the Matriarch Beata, or Wesley's smug little smirk, these are characters existing in a state of high self-regard. Of course, this is one of the most common complaints about the beginnings of the series, how the characters are bland and complacent to a fault. To the show's credit, that changes and there are some entertaining challenges to the crew's arrogance later on.
A central motif in the Star Trek universe is an attraction to competency. Although flawed, the members of the crew, from the earliest episodes, clearly represent the best of the best. Picard is the premier Star Fleet captain and Riker is an ambitious and impressive first officer. Data, LaForge, and Worf are all unique individuals with, presumably, glowing resumes. Perhaps that's why the crew comes off as remarkably intolerant in some of the earliest episodes: repulsed by the Ferengi, patronizing to the Ligonians, and dismissive of Angel matriarchal oligarchy. The two basic types of threats in these earlier stories are either godlike in potency or significantly more primitive. There's not much for the crew to test their mettle against.
For the first season then, most of the conflicts revolve around the one character still attempting to define his abilities, Wesley Crusher. This is a tricky statement because Wesley also fulfills a more odious function in the first season: a convenient deux ex machina. Is an enormous gob of glowing star barf heading towards the Enterprise? Wesley fixes it. Is the ship marooned in a swirling, gossamer fantasy mist at the end of the universe? Wesley fixes it. The list of examples where Wesley saves the day at the last minute is depressingly long. But that's not really an example of an ego conflict. A better moment happens in the episode "Coming of Age." Here Wesley is attempting the Star Fleet admission test, competing against three other applicants, all demonstrating impressive amounts of knowledge and talent. One section of the test, challenges the applicants to solve a series of engineering problems. One of Wesley's competitors, Mordock, loses composure and Wesley, perhaps empathizing with the other's struggles, talks him through to a solution. The situation is sort of obvious, but I like it nevertheless as an example of ego played against a need for acceptance. From an earlier encounter between Wesley and Jack, we see Wesley feels uncomfortable with his talent, embarrassed by outcompeting his friend. So again he is confronted with the tension of surging ahead for himself or pausing to help someone else. He helps his new friend, allowing Mordock to win, and losing a few crucial points.
I think this is the same basic conflict that appears in the psych test later in the episode. Here, Wesley is confronted with an apparent explosion at the test site. He rushes in, attempting to bring two trapped personnel to safety. He is forced to physically drag one free from an impending explosion, and is unable to convince the other to leap through a jet of noxious gases to safety. Ultimately Wesley chooses to save the injured man, leaving the other to his fate. Informed that the whole situation was the psych test, Wesley shows remarkable self-control. Instead of grabbing the nearest phaser and blasting his way out of the Skinner box, Wesley realizes his central problem is a fear he could not choose one life over another or withstand the pressures of responsibility. He fails to win his way into Star Fleet, but the test has given him some measure of confidence in his own abilities.
I honestly feel this is one type of conflict under-used in speculative fiction. Skill and talent is typically taken for granted in stories. An uber-competent hero cuts a wide swath through swarms of cannon fodder until he meets his only one true challenge in a story, the principal villain. The rivals match swords, guns, wits or giant mechs in a pitched battle until one of them wins, roll credits. I'd like more examples of conflicts where a competent character has a central talent truly tested in a convincing and interesting way. Perhaps, that conflict has to look like what I've described above, the need to compete versus a need to help others.