The best upper bound is $1$, no matter what the values of $R_1^2$ and $R_2^2$ may be.
The following discussion explains why, in three increasingly detailed ways. The first explanation gives geometric intuition, leading to a simple example. The second one translates that into a procedure to generate specific datasets that give rise to this example. The third one generalizes this procedure to show how any mathematically possible value of $R^2$ can be achieved, given arbitrary values of $R_1^2$ and $R_2^2$.
I adopt a notation in which the independent variables are named $x_1$ and $x_2$ (rather than $x$ and $z$), so that the distinction between the independent and dependent variables remains clear.
(A comment by the alert @f coppens compels me to add that these results change when one or more of the regressions does not include a constant term, because then the relationship between $R^2$ and the correlation coefficients changes. The methods used to obtain these results continue to work. Interested readers may enjoy deriving a more general answer for that situation.)
For the simple regressions (1) and (2), the $R_i^2$ are the squares of the correlation coefficients between $x_i$ and $y$. Relationships among correlation coefficients are just angular relationships among unit vectors in disguise, because the correlation coefficient of two variables $x$ and $y$ (considered as column $n$-vectors) is the dot product of their normalized (unit-length) versions, which in turn is the cosine of the angle between them.
In these geometric terms, the question asks
How close can a vector $y$ come to the plane generated by $x_1$ and $x_2$, given the angles between $y$ and the $x_i$?
Evidently $y$ can actually be in that plane, provided you put $y$ at a given angle $\theta_1$ with $x_1$ and then place $x_2$ at a given angle $\theta_2$ with $y$. When that happens, the $R^2$ for regression (3) is $1$, demonstrating there is no meaningful upper bound on $R^2$.
Geometric thinking is no longer considered rigorous, but it leads us to a rigorous example. Start with two orthogonal unit vectors $u$ and $v$, each of which is orthogonal to a vector of ones (so that we can accommodate a constant term in all three regressions). Given $R_1^2$ and $R_2^2$, let $\rho_i^2 = R_i^2$ be choices of their square roots. To place vectors $x_1$, $y$, and $x_2$ at the required angles, set
$$\eqalign{
&x_1 &= u\\&y&=\rho_1 u + \sqrt{1-\rho_1^2} v\\ &x_2 &= (\rho_1\rho_2-\sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2})u + (\rho_1\sqrt{1-\rho_2^2}\sqrt{1-\rho_1^2})v.}$$
Since $u\cdot u = v\cdot v = 1$ and $u\cdot v = 0$, you can verify that $x_2\cdot x_2 = 1$ as required,
$$y\cdot x_1 = \rho_1,$$
and
$$\eqalign{
y\cdot x_2 &= \rho_1\left(\rho_1\rho_2-\sqrt{1-\rho_1^2}\sqrt{1-\rho_2^2}\right) + \sqrt{1-\rho_1^2}\left(\rho_1\sqrt{1-\rho_2^2}\sqrt{1-\rho_1^2}\right) \\
&= \rho_2,}$$
as intended.
For a completely concrete example with $n\ge 3$ observations, start with any two $n$-vectors $u_0$ and $v_0$ which are linearly independent and linearly independent of the $n$-vector $\mathbf{1}=(1,1\ldots, 1)$. Apply the Gram-Schmidt process to the sequence $\mathbf{1}, u_0, v_0$ to produce an orthonormal basis $\mathbf{1}/\sqrt{n}, u, v$. Use the $u$ and $v$ that result. For instance, for $n=3$ you might start with $u_0 = (1,0,0)$ and $v_0=(0,1,0)$. The Gram-Schmidt orthogonalization of them yields $u = (2,-1,-1)/\sqrt{6}$ and $v=(0,1,-1)/\sqrt{2})$. Apply the preceding formulas to these for any given $R_1^2$ and $R_2^2$ you desire. This will result in a dataset consisting of the $3$-vectors $x_1$, $x_2$, and $y$ with the specified values of $R_1^2, R_2^2$, and $R^2 = 1$.
A similar approach, starting with mutually orthonormal vectors $u_0, v_0, w_0$, can be used to construct examples in which $R^2$ achieves any specified value in the interval $[\max(R_1^2, R_2^2), 1]$. Order the $x_i$ so that $R_1^2 \ge R_2^2$. Writing $y = \alpha u_0 + \beta v_0 + \gamma w_0$, $x_1 = u_0$, and $x_2 = \rho_{12}u_0 + \sqrt{1-\rho_{12}^2}v_0$, compute that $\rho_1 = \alpha$ and $\rho_2 = \alpha \rho_{12} + \beta \sqrt{1-\rho_{12}^2}$. From this, and the fact that $\alpha^2+\beta^2+\gamma^2=1$, solve and find that
$$\beta = \frac{\rho_2 - \rho_1\rho_{12}}{\sqrt{1-\rho_{12}^2}}$$
and $\gamma = \sqrt{1-\alpha^2 - \beta^2}$. For this square root to exist, $\beta$ needs to be small, but that can be guaranteed by choosing $\rho_{12}$ (the correlation between the two independent variables $x_1$ and $x_2$) to be small in size, because as $\rho_{12}$ approaches $\rho_2/\rho_1$, (which is possible because the absolute value of this ratio does not exceed $1$), $\beta$ approaches zero continuously.
The cognoscenti will recognize the relationship between the formula for $\beta$ and a certain partial correlation coefficient.
