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?



Examples for open disc bundle which is not vector bundle
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 $20$. 


