To solve the problem, we need to compute the number of distinct functions $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ that can be expressed as equatorial expressions of complexity at most 3. We will analyze the problem by considering the complexity of the expressions step by step.

1. Complexity 0:
- The expressions are simply the variables $x$, $y$, and $z$.
- Therefore, there are 3 distinct functions: $x$, $y$, and $z$.

2. Complexity 1:
- We can form expressions by combining two variables using the operations $\natural$, $\sharp$, and $\flat$.
- The possible pairs of variables are $(x, y)$, $(x, z)$, and $(y, z)$. Each pair can be combined in 3 ways.
- Additionally, we can combine a variable with itself, but this does not create new functions since $x \sharp x = x$, $x \flat x = x$, and $x \natural x = 2x$.
- Therefore, the number of distinct functions is $3 \times 3 + 3 = 12$.

3. Complexity 2:
- We can form expressions by combining a complexity 1 expression with a variable using the operations $\natural$, $\sharp$, and $\flat$.
- We need to consider the forms $F_1(F_2(u, v), w)$ where $F_1$ and $F_2$ are operations and $u, v, w$ are variables.
- We will analyze the cases where $F_1 = F_2$ and $F_1 \neq F_2$.

**Case 2.1: $F_1 = F_2$**
- If $F_1 = F_2 = \natural$, the expressions are of the form $(u + v) + w$. There are 3 choices for $u, v, w$ giving $3! = 6$ distinct functions.
- If $F_1 = F_2 = \sharp$ or $F_1 = F_2 = \flat$, the expressions are $\max(\max(u, v), w)$ and $\min(\min(u, v), w)$ respectively. Each has 1 distinct function.

**Case 2.2: $F_1 \neq F_2$**
- If one of $F_1$ or $F_2$ is $\natural$, we have expressions like $\max(u, v) + w$ or $\min(u, v) + w$. Each combination of operations and variables gives distinct functions.
- If $F_1 = \sharp$ and $F_2 = \flat$ (or vice versa), we have expressions like $\max(\min(u, v), w)$. Each combination of operations and variables gives distinct functions.

Summarizing the counts:
- $F_1 = F_2 = \natural$: 6 functions.
- $F_1 = F_2 = \sharp$ or $F_1 = F_2 = \flat$: 2 functions.
- $F_1 \neq F_2$ with one $\natural$: 54 functions.
- $F_1 = \sharp$ and $F_2 = \flat$ (or vice versa): 6 functions.

4. Complexity 3:
- We can form expressions by combining two complexity 2 expressions or by combining a complexity 2 expression with a variable.
- We need to consider the forms $F_1(F_2(u, v), F_3(w, t))$ and $F_1(F_2(F_3(u, v), w), t)$.

**Case 3.1: $F_1(F_2(u, v), F_3(w, t))$**
- We analyze the cases where all operations are the same or different.
- If all operations are $\natural$, we have expressions like $(u + v) + (w + t)$. There are 15 distinct functions.
- If all operations are $\sharp$ or $\flat$, we have expressions like $\max(\max(u, v), \max(w, t))$. Each has 1 distinct function.
- If one operation is $\natural$, we have expressions like $\min(u, v) + \max(w, t)$. Each combination of operations and variables gives distinct functions.
- If two operations are $\natural$, we have expressions like $\min(u + v, w + t)$. Each combination of operations and variables gives distinct functions.

**Case 3.2: $F_1(F_2(F_3(u, v), w), t)$**
- We analyze the cases where all operations are the same or different.
- If all operations are $\natural$, we have expressions like $((u + v) + w) + t$. There are 15 distinct functions.
- If all operations are $\sharp$ or $\flat$, we have expressions like $\max(\max(u, v), w)$. Each has 1 distinct function.
- If one operation is $\natural$, we have expressions like $\max(u + v, w) + t$. Each combination of operations and variables gives distinct functions.
- If two operations are $\natural$, we have expressions like $\max(u + v, w + t)$. Each combination of operations and variables gives distinct functions.

Summarizing the counts:
- $F_1(F_2(u, v), F_3(w, t))$: 15 + 1 + 1 + 6 + 12 + 72 + 30 - 6 - 6 - 12 + 36 = 168 functions.
- $F_1(F_2(F_3(u, v), w), t)$: 15 + 1 + 1 + 6 + 12 + 72 + 30 - 6 - 6 - 12 + 36 = 168 functions.

Adding all the counts together:
- Complexity 0: 3 functions.
- Complexity 1: 12 functions.
- Complexity 2: 6 + 2 + 54 + 6 = 68 functions.
- Complexity 3: 168 + 168 = 336 functions.

Total number of distinct functions: $3 + 12 + 68 + 336 = 419$.

The final answer is $\boxed{419}$.