  We study certain properties of six-dimensional tensionless E-strings (arising
from zero size $E_8$ instantons). In particular we show that $n$ E-strings form
a bound string which carries an $E_8$ level $n$ current algebra as well as a
left-over conformal system with $c=12n-4-{248n \over n+30}$, whose characters
can be computed. Moreover we show that the characters of the $n$-string bound
state are captured by N=4 U(n) topological Yang-Mills theory on $\half K3$.
This relation not only illuminates certain aspects of E-strings but can also be
used to shed light on the properties of N=4 topological Yang-Mills theories on
manifolds with $b_2^+=1$. In particular the E-string partition functions, which
can be computed using local mirror symmetry on a Calabi-Yau three-fold, give
the Euler characteristics of the Yang-Mills instanton moduli space on $\half
K3$. Moreover, the partition functions are determined by a gap condition
combined with a simple recurrence relation which has its origins in a
holomorphic anomaly that has been conjectured to exist for N=4 topological
Yang-Mills on manifolds with $b_2^+=1$ and is also related to the holomorphic
anomaly for higher genus topological strings on Calabi-Yau threefolds.
