As $m \in \mathbb{Z}^{+}$ gets bigger and bigger, the unique positive real root of the equation ${^{m + 1}x} = {^{m}x} + 1$, where ${^{m}x} := x^{x^{x^{\dots}}} \}m\text{-many} \; x\text{'s}$, gets progressively smaller.
Now, since $x > 1$ trivially holds for any $m$ (i.e., $1^{1^{\dots}}=1$ so that $1 \neq 1+1$), it would be interesting to find the convergence value of the serie $x : {^{2}x} = {^{1}x} + 1, x : {^{3}x} = {^{2}x} + 1, x : {^{4}x} = {^{3}x} + 1, \dots, x : {^{m + 1}x} = {^{m}x} + 1$ as $m$ approaches infinity.
In details, we have that
if $m = 1$, then $1.7767750 < x(m) < 1.7767751$;
if $m = 2$, then $1.6712921 < x(m) < 1.6712922$;
$\dots$
if $m = 11$, then $1.4837622 < x(m) < 1.4837623$;
and so forth.
Moreover, since it has been proved that $x : {^{2}x} = {^{1}x} + 1$ is transcendental (see Gelfond-Schneider Theorem), can we say the same thing for any $x(m : m \geq 2)$?