Let \(k\) be a finite field with \(q\) elements. So far there are several interesting examples of asymptotically good towers of algebraic function fields \(\mathcal F: F_0\subseteq F_1\subseteq\ldots\) which are defined by a polynomial of type \(f(x,y)=g_1(x)h(y)-g_0(x)\), \(h(y)\) being an additive polynomial, in the sense that: \(F_0:=k(x_0)\), \(F_i:=F_{i-1}(x_i)\), \(f(x_{i-1},x_i)=0\) for \(i\geq 1\); see e.g. [\textit{A. Garcia} and \textit{H. Stichtenoth}, Invent. Math. 121, No. 1, 211--222 (1995; Zbl 0822.11078), J. Number Theory 61, No. 2, 248--273 (1996; Zbl 0893.11047); \textit{G. van der Geer} and \textit{M. van der Vlugt}, Bull. Lond. Math. Soc. 34, No. 3, 291--300 (2002; Zbl 1062.11037)]. Let \(h(y)=y^p+by\) and \(G(x):=g_0(x)/g_1(x)\). If \(\mathcal F\) is asymptotically good, then \(G(x)\) is one of the following types: (1) \(G(x)=(x-c)^p/G_1(x)+a\), (2) \(G(x)=G_1(x)/(x-a)^p\), (3) \(G(x)=1/G_1(x)+a\) [\textit{P. Beelen, A. Garcia} and \textit{H. Stichtenoth}, Bull. Braz. Math. Soc. 35, No. 2, 151--164 (2004; Zbl 1119.14022)]. The aforementioned examples are all of type (1) and it is an open problem the existence of asymptotically good towers of type (2) and (3). In this paper the authors construct a tower of type (3) whose genus, namely \(\lim_{i\to \infty}g(F_i)/[F_i:F_0]\) is finite; this is a necessary condition in order the tower be asymptotically good.