= Solution
For each integer $m\geq1$, list the $m^2$ <functions>
$$
f_{m,j}(x)=m\,\mathbf1_{[j/m^2,(j+1)/m^2)}(x),\qquad0\leq j<m^2,
$$
in increasing $j$, and concatenate the blocks in increasing $m$. Each <function> has $L^1$ norm $1/m$, so the resulting <sequence> has norm tending to zero. Every $x\in(0,1)$ lies in exactly one interval of each block, where the <function> takes value $m$. Therefore
$$
\boxed{\|f_n\|_1\to0,\qquad\limsup_n f_n(x)=\infty\quad(0<x<1).}
$$
This <unbounded typewriter sequence with vanishing L1 norm> is a moving-spike construction: the support shrinks fast enough to beat the increasing height.
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