How to make a flex item not fill the height of the flex container?

The align-items, or respectively align-content attribute controls this behaviour.

align-items defines the items' positioning perpendicularly to flex-direction.

The default flex-direction is row, therfore vertical placement can be controlled with align-items.

There is also the align-self attribute to control the alignment on a per item basis.

#a {
  display:flex;

  align-items:flex-start;
  align-content:flex-start;
  }

#a > div {
  
  background-color:red;
  padding:5px;
  margin:2px;
  }
 #a > #c {
  align-self:stretch;
 }

<div id="a">
  
  <div id="b">left</div>
  <div id="c">middle</div>
  <div>right<br>right<br>right<br>right<br>right<br></div>
  
</div>

css-tricks has an excellent article on the topic. I recommend reading it a couple of times.

When you create a flex container various default flex rules come into play.

Two of these default rules are flex-direction: row and align-items: stretch. This means that flex items will automatically align in a single row, and each item will fill the height of the container.

If you don't want flex items to stretch – i.e., like you wrote:

make its height the minimum required for holding its content

... then simply override the default with align-items: flex-start.

#a {
  display: flex;
  align-items: flex-start; /* NEW */
}
#a > div {
  background-color: red;
  padding: 5px;
  margin: 2px;
}
#b {
  height: auto;
}

<div id="a">
  <div id="b">left</div>
  <div>
    right<br>right<br>right<br>right<br>right<br>
  </div>
</div>

Here's an illustration from the flexbox spec that highlights the five values for align-items and how they position flex items within the container. As mentioned before, stretch is the default value.

^{Source: W3C}