Is it possible to have a simile that does not contain the words 'like' or 'as'?
Would the following sentence exemplify such a simile?:
"He was handsome in a way that required a bit of work from the viewer."
Or is this a metaphor?
If you are simply looking for a way to express that something is like something else in some way, without using the word like or as, I think Jim's comment from a couple of months ago is on point. For example, instead of saying
He was as wild as the west wind.
you could say
He was wild in the same way that the west wind is wild.
or (somewhat differently)
His wildness was the wildness of the west wind.
In neither case is the reader likely to take literally the assertion that a person's wildness and a wind's wildness are really the same. Instead the reader will be inclined to read the asserted sameness as a simile-like likeness.
Similarly, if you reword
He was running around like a chicken with its head cut off.
He was running around the way a chicken with its head cut off might have done.
you express the same idea of similarity without using like or as.
Usually, the like or as form is less long-winded and thus more appealing to the reader's or listener's ear because it's more concise. But if the task is to express the substance of a simile without using like or as, it can certainly be done by using some form "in the same way that" or "the way."
Having said that, I agree with John Feltz's comment that
He was handsome in a way that required a bit of work from the viewer.
is neither a simile nor a metaphor. And neither is
He was as handsome as he could be, given that he depended on the lively imagination of others to make him seem attractive.