Unary Turing-unrecognizable language
Suppose that you have an algorithm $f$ that decides $L=\{1^{BB(k)}\mid k \in \mathbb N\}\subseteq \{1^n\mid n\in \mathbb N\}$. Then you can build the following algorithm that computes $BB$:

// input k
n = 0
values_of_bb = []
while length(values_of_bb) &lt; k
  if f(1^n) == 1
    values_of_bb.push(n)
    n = n + 1
return values_of_bb[k]


This will be true for all strictly increasing functions from $\mathbb N$ to $\mathbb N$: if the image is decidable, then the function is computable. You just have to enumerate potential values in increasing order and put them in a list (still in increasing order), and then the $k^\text{th}$ value is simply the $k^\text{th}$ element of this list.