Q:

Examples for open disc bundle which is not vector bundle

William Browder showed in "Open and closed disc bundles", Ann. of Math. (2) 83 (1966), 218-230 that there are open disc bundles over some complex which cannot be isomorphic to a vector bundle.
My question is: Is there any related examples?
For example: What is the least dimensions for the base complex and the dimension of the fiber?
Or is there any explicit example for this complex?

A:

Given a disc bundle over a space $X$, there's the classifying map 
$$ X \to BDiff(D^n)$$
$Diff(D^n)$ has as a subgroup $O_n$, the orthogonal group.  The disc bundle over $X$ is a vector bundle if and only if the classifying map $X \to BDiff(D^n)$ factors (up to homotopy) through a map $X \to BO_n$, where the other map is the induced map $BO_n \to BDiff(D^n)$ induced from inclusion $O_n \to Diff(D^n)$. 
There's a homotopy fiber sequence
$$PDiff(D^n) \to Diff(D^n) \to O_n$$
where $PDiff(D^n)$ is the subgroup of diffeomorphisms of $D^n$ which restrict to the identity on a hemi-disc. The idea is to think about restricting a diffeomorphism of $D^n$ to a half of the disc, and linearizing that embedding. The fiber is sometimes called the pseudo-isotopy diffeomorphism group of $D^{n-1}$.  Moreover, this fiber sequence is a direct product (up to homotopy)
$$Diff(D^n) \times O_n \simeq PDiff(D^n)$$
So your question boils down to the existence of non-null maps $X \to B(PDiff(D^n))$. 
The homotopy type of the pseudo-isotopy diffeomorphism group isn't terribly well known but there are things known about it.   Cerf's theorem says $PDiff(D^n)$ is connected for $n$ large.  A big result in this area is Kiyoshi Igusa's stability theorem, which tells you that certain stabilization maps $PDiff(D^n) \to PDiff(D^{n+1})$ induce isomorphisms on a range of homotopy groups, and the stable pseudo-isotopy diffeomorphism group is then relateable to K-theory.   I don't know what the state of the art is in computing those homotopy groups, but people like John Rognes probably know. 
$\pi_1 PDiff(D^n)$ has four elements for $n$ large. This is in Hatcher's 1978 Concordance spaces, higher himple homotopy theory and applications Proc. Symp. Pure. Math. 32.  I haven't chased through the details but I believe "n large" in Hatcher's paper means Kiyoshi Igusa's stable range, which in this case I believe it means $n \geq 11$. 
So your base space need only by $S^2$ provided the fiber dimension is high enough. Hatcher gives an explicit construction of related homotopy-classes in his paper although it's a fair bit of work to digest it.  
edit: Looking at the Browder paper it appears you're interested in topological bundles where the fiber is a non-compact ball.   In this case you're interested in the lifting property $X \to BHomeo(\mathbb R^n)$  for the map $BO_n \to BHomeo(\mathbb R^n)$.  
There's quite a bit known about the inclusion $O_n \to Homeo(\mathbb R^n)$.   The Cerf-Morlet Comparison Theorem says $Diff(D^n, fix \partial D^n)$ has the homotopy-type of $\Omega^{n+1}(PL_n/O_n)$.  This says that diffeomorphisms of $D^n$ that fix the boundary in some sense measure the existence of $PL$ bundles that are not vector bundles.   There are similar statements for topological bundles.  There are non-trivial elements in the homotopy of $Diff(D^n fix \partial D^n)$ due to people like Igusa, Hatcher and Farrell.  This would tell you that there are PL bundles with fiber $\mathbb R^n$ which are not smooth bundles.  In this case the dimension of your base sphere would have to be larger, but order-of-magnitude a little less than $20$. 

A:

Some remarks which amount in some way to an answer.
(1) Let  $\text{Diff}(D^n)$ be the diffeomorphisms of $D^n$ which restrict to the identity on
the boundary.
When $n\gg k$ is large, $\pi_k(\text{Diff}(D^n))\otimes \Bbb Q$ was computed
by Farrell and Hsiang. The answer is that this is $\cong \Bbb Q$ when $k = 4j-1$ and $n$ is odd. In every other case this group is trivial.
(2) Let $\text{Diff}(D^n,S^{n-1})$ be the group of diffeomorphisms of $D^n$ which aren't necessarily the identity on the boundary of $D^n$. 
Then as in Ryan's answer there's a homotopy fiber
sequence
$$
C(S^{n-1}) \to \text{Diff}(D^n,S^{n-1}) \to O(n)
$$
where $C(S^{n-1})$ is the concordance space of the $(n-1)$-sphere. This is the group of
diffeomorphisms of $S^{n-1} \times [0,1]$ which restrict to the identity on 
$S^{n-1} \times 0$.
As Ryan notes
$$
\pi_k(\text{Diff}(D^n,S^{n-1})) \cong \pi_k(C(S^{n-1})) \times  \pi_k(O(n)) .
$$
Again, when $n\gg k$, we can compute $\pi_k(O(n)) \cong \pi_i(O)$ by Bott periodicity.
Rationally, it's given by $\Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$
and trivial otherwise.
Also $ \pi_k(C(S^{n-1}))\otimes \Bbb Q$ can be computed when $n \gg k$. It's also 
$\cong \Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$ and trivial otherwise.
So we get, for $n \gg k$, that $\pi_k(\text{Diff}(D^n,S^{n-1})) \otimes \Bbb Q$ is
isomorphic to $\Bbb Q \oplus \Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$ and trivial otherwise.

