Order statistics for non i.i.d. variables $u$ is a random variable with uniform distribution in $[a,b]$ and is not observed (the distribution is known though). At every period $t \in \{2, ..., T\}$, the researcher observes $Y_{t+1} = \alpha Y_t + (1 - \alpha) u_t$, where $u_t$ is an i.i.d. random draw from the uniform distribution in $[a,b]$. $T$, $\alpha$ and $Y_1$ are known. 
How to find the distribution of the highest order statistic of $Y_t$ at every period $t$?
--- EDIT ---
I have the following derivations so far. Assume that we are at period 1:
(1) Rewrite $Y_t$ as:
\begin{aligned}
Y_t  &= \alpha Y_{t -1 } + (1 - \alpha) u_{t-1} \\
     &= \alpha^{(t-1)} Y_1 + (1 - \alpha) \sum_{j=1}^{t} u_{t-j} \alpha^{j-1}
\end{aligned}
Similarly for period 2
\begin{aligned}
Y_t  &= \alpha Y_{t -1 } + (1 - \alpha) u_{t-1} \\
     &= \alpha^{(t-2)} Y_2 + (1 - \alpha) \sum_{j=2}^{t} u_{t-j} \alpha^{j-2}
\end{aligned}
 A: Your problem features not only non-identical distributions, but also dependence. It would be difficult enough to find a solution given numerical T (e.g. T = 10), never mind a general symbolic solution or dependence.
Nevertheless, there is always Monte Carlo ... here is the simulated pdf of the sample maximum when $T  = 10$ (top diagram) and $T = 50$ (bottom diagram). In all cases, $Y_1 = \frac12$, with $\alpha = \frac18, \frac12, \frac78$, as noted. The dashed line denotes a standard Uniform, for comparison.

(source: tri.org.au)
The limit distributions when $\alpha = 0$ and 1 are easy to derive, and make a helpful frame of reference.
A: Not really an answer to this quite difficult question, but too long
for a comment.
I doubt that a usable explicit form of the distribution can be found,
and the Monte-Carlo illustrated by @wolfies seems the only viable
approach. A related and less difficult question arises from the
stationary case, when the initial $Y_0$ is assumed to be drawn from
the stationary distribution: then for large $t$, how can we compute
the distribution of $M_t := \max \{Y_1, \, \dots, \,Y_t\}$?
Extreme Value Theory (EVT) is most known for the i.i.d. context, and the
Fisher-Tippet-Gnedencko theorem
gives the possible limit distributions for $M_t$. 
When $\alpha = 0$ we know from EVT that $V_t:= t [M_t - 1]$ converges
in distribution to the reverse exponential with density $e^v 1_{v \leq
0}(v)$.
Extremes of stationary (dependent) sequences have also been studied
quite extensively, and form an important chapter of EVT. The book by
Embrechts P., Klüppelberg C. and Mikosch T. is a very good resource on EVT and
includes several sections (notably 4.4, 5.5 and 8.1) in relation with
the question. For a stationary sequence $Y_t$, the asymptotic
distribution (for large $t$) of the maximum $M_t$ can be related to
the marginal distribution in some cases. A number of results are
available under mixing conditions: then the distribution of the max
of the $t$ dependent observations is for large $t$ the same as the
distribution of $t \theta$ i.i.d. r.vs with the same distribution,
where $0 < \theta \leq 1$ is the extremal index and relates to the
(extremal) dependence between the r.vs $Y_t$.  For a gaussian AR(1)
as well as for stationary gaussian ARMA, if is known that $\theta =
1$, so these processes have (asymptotically) the same extremal
behaviour as an i.i.d. gaussian sequence.
In your case, for $0 < \alpha < 1$ the stationary (marginal)
distribution is not known explicitly but could be investigated since
an expression can be given for the characteristic function or Laplace
transform see Andel and
Hrach.  The
stationary distribution has support $[0,\,1]$ and has
a density which is infinitely derivable, including at its upper
end-point where all derivatives must vanish. This behaviour favours
the conjecture that the stationary distribution is in the Gumbel
domain of attraction, and not in the reverse Weibull domain as was the
case for $\alpha = 0$. I believe that the Von Mises conditions hold,
but did not yet succeed at proving it.
It could be a good approach to begin by finding the domain of
attraction of the stationary distribution, by investigating the
behaviour of the density near the finite upper end-point. Then see if
a mixing condition holds (which seems likely), and then find the
extremal index $\theta$. This might be a material for other 
questions.
