Let's write $\omega \equiv \lambda y. y \: y$.
So your term is $(\lambda x. (\lambda y . y \: x)(\lambda z . x \: z))\: \omega$.
We know that leftmost reduction will always find the $\beta$-normal form, so let's do that. The reduction sequence we get is:
\begin{align*}
(\lambda x. (\lambda y . y \: x)(\lambda z . x \: z))\: \omega &\to_{\beta} (\lambda y . y \: \omega)(\lambda z . \omega \: z)\\
&\to_{\beta} (\lambda z . \omega \: z) \: \omega \\
&\to_{\beta} \omega \: \omega
\end{align*}
Now what happens when we reduce $\omega \: \omega \equiv (\lambda y. y \: y) \: \omega$?
There's only one reduction we can make, and it's $(\lambda y. y \: y) \: \omega \to_{\beta} (y \: y)[\omega/y] \equiv \omega \: \omega$.
So the term $\omega \: \omega$ only reduces to itself! Now it's clear that we can keep repeating this reduction forever, so our original term has no normal form.
The term $\omega \: \omega$ is known as $\boldsymbol{\Omega}$, and it's one of the simplest terms with no $\beta$-normal form.
Depending on the definition of leftmost, the adjective outermost may be or not be superfluous.
The most common meaning of leftmost is the one pointed out by Lemontree in his comment, and it refers to the syntactic tree associated with each $\lambda$-term. First, you define the notion of outermost and innermost redexes.
In a $\lambda$-term $M$, a redex is
- outermost if it is not a subterm of any other redex in $M$;
- innermost if it does not contain any other redexes as subterms.
A term may contain several outermost redexes and several innermost redexes. For instance, in the $\lambda$-term $x ((\lambda y.y)y) ((\lambda z.z)z)$, the two redexes are outermost (and innermost too!). Two fix an order so as to define an evaluation strategy, we choose the leftmost or the rightmost redex among the outermost or innermost ones, where leftmost (resp. rightmost) means the one that occurs leftmost (resp. rightmost) in the syntax tree of the $\lambda$-term. In this way, there is at most one leftmost-outermost redex in a $\lambda$-term, and similarly for leftmost-innermost, rightmost-innermost, etc.
Clearly, this definition of leftmost (and of rightmost) makes sense only together with the notion of outermost or innermost.
A second possible meaning of leftmost is less common but it is used for instance in Krivine's textbook. Since a redex has necessarily the form $(\lambda x.M)N$, in every $\lambda$-term, each subsequence of the form "$(\lambda$" corresponds to a unique redex in that term. We can then define the leftmost redex in a $\lambda$-term $M$ as the redex whose "$(\lambda$" part occurs leftmost in $M$, seen as a string of characters.
In this second definition of leftmost, the adjective outermost is clearly superfluous: the leftmost redex (in this second meaning) is necessarily outermost (in the sense defined above). Actually, this second meaning of leftmost coincides with the definition of leftmost-outermost above: both definitions identify the same redex in a $\lambda$-term (if any), although they follow a different procedure to identify it. Indeed, the only difference is in the meaning of the word "leftmost" in the definitions of "leftmost-outermost redex" and of "leftmost redex".
In the literature, the definition of the normal order reduction via the notion leftmost-outermost redex is common (and often preferred to the one via the second meaning of leftmost) because it is flexible: indeed, using the same approach one can define other evaluation strategies based on the leftmost-innermost redex, the rightmost-innermost redex, etc.
Best Answer
Basically, the function $(\lambda f.\lambda x.f(fx))$ is applied to the argument $\lambda y.y+1$. This step is also called beta reduction.