Let us analyze the process step by step.

Let $C_n$ be the circumference after $n$ minutes.

At the start ($n = 0$):
The figure is a square with side $1$, so the circumference is $4 \times 1 = 4$.

Minute 1:
On each vertical side, a square of side $\frac{1}{3}$ is added in the middle. Each such square adds $2$ new vertical segments of length $\frac{1}{3}$ (the left and right sides of the small square), and replaces a segment of length $\frac{1}{3}$ on the original side with the top and bottom of the small square (which are also length $\frac{1}{3}$ each). The total length added is $4 \times 4 \times \frac{1}{3}$ (since $4$ squares are added, each with $4$ sides of length $\frac{1}{3}$), but we must subtract the length that is replaced (the $\frac{1}{3}$ segments on the original sides, $2$ per square, so $4 \times 2 \times \frac{1}{3}$).

But let's look for a pattern.

Let us focus on the vertical segments:
- At step $0$, there are $2$ vertical segments (the left and right sides of the square), each of length $1$.
- At step $1$, on each vertical segment, a square is added in the middle, so each vertical segment is split into $2$ segments of length $\frac{1}{3}$ (the parts not covered by the new square), and $2$ vertical sides of the new square (each $\frac{1}{3}$).

But the key is that at each step, every vertical segment is replaced by $2$ vertical segments, each $\frac{1}{3}$ the length of the previous segment.

Let us formalize this:
Let $V_n$ be the number of vertical segments after $n$ steps, and $L_n$ be the length of each vertical segment.

At $n = 0$: $V_0 = 2$, $L_0 = 1$
At $n = 1$: $V_1 = 4$, $L_1 = \frac{1}{3}$
At $n = 2$: $V_2 = 8$, $L_2 = \frac{1}{9}$
In general: $V_n = 2^{n+1}$, $L_n = \frac{1}{3^n}$

The total length of vertical segments at step $n$ is $V_n \times L_n = 2^{n+1} \times \frac{1}{3^n}$

Now, for the horizontal segments:
At each step, the number of horizontal segments increases as well, but the process is similar.

But notice that the total circumference at each step is:
- At $n = 0$: $4$
- At $n = 1$: $4 + 4 \times (\frac{4}{3} - \frac{2}{3}) = 4 + 4 \times \frac{2}{3} = 4 + \frac{8}{3}$
But let's use the recursive pattern.

Alternatively, notice that at each step, the total added length is $2^{n+1} \times \frac{2}{3^n}$ (since each new square adds $4$ sides of length $\frac{1}{3^n}$, but $2$ of these replace existing segments).

But the process is self-similar, and the total circumference at step $n$ is:

$C_n = 4 \left(1 + \frac{2}{3} + \frac{2}{3^2} + \cdots + \frac{2}{3^{n}}\right)$

This is a geometric series:

$C_n = 4 \left(1 + 2 \sum_{k=1}^{n} \frac{1}{3^k}\right)$

The sum $\sum_{k=1}^{n} \frac{1}{3^k} = \frac{1/3 (1 - (1/3)^n)}{1 - 1/3} = \frac{1}{2} (1 - (1/3)^n)$

So:
$C_n = 4 \left(1 + 2 \times \frac{1}{2} (1 - (1/3)^n)\right) = 4 (1 + 1 - (1/3)^n) = 4 (2 - (1/3)^n) = 8 - 4 (1/3)^n$

After $60$ minutes, $n = 60$:

$C_{60} = 8 - 4 \left(\frac{1}{3^{60}}\right)$

Since $\frac{1}{3^{60}}$ is extremely small, the circumference is very close to $8$.

Answer:

The circumference of the figure after one hour is $8 - 4 \left(\frac{1}{3^{60}}\right)$.