We are tasked with determining whether it is possible to arrange the numbers $1^1, 2^2, \ldots, 2008^{2008}$ in such a way that the resulting sequence forms a perfect square.

To address this question, let's analyze the problem step-by-step.

### Overview and Initial Considerations

1. Understand the Range of Numbers: The numbers given are powers, specifically $k^k$ for integers $k$ ranging from 1 to 2008. Each is an integer power raised to itself.

2. Structure the Problem: The goal is to arrange these constructed exponential numbers $1^1, 2^2, \ldots, 2008^{2008}$ in a sequence such that their concatenation results in a perfect square.

### Concatenation and Properties

1. Size of the Number: Consider the magnitude of numbers like $2008^{2008}$, which is a very large number. The length of these numbers varies significantly as $k$ increases, given that each is the power of the integer to itself.

2. Length Consideration for Concatenation: When concatenating such numbers, the resulting integer is tremendously large, far exceeding the manageable size for perfect squares that exhibit regular digit patterns.

### Analysis of Perfect Square Properties

1. Perfect Square Modulo Analysis: Numbers that are perfect squares exhibit particular patterns when considered modulo small integers. For instance, when divided by 3, a perfect square leaves a remainder of either 0 or 1. Whereas, modulo 4, a perfect square leaves a remainder of 0 or 1, and so on.

2. Sum of Digits Property: The sum of the digits of a perfect square cannot match certain specific outcomes when studied modulo 9. Large concatenated numbers structured like this often do not satisfy divisibility conditions that perfect squares do satisfy.

3. Arrangement Incongruities:
- When the digits of $1^1, 2^2, \ldots, 2008^{2008}$ are randomly arranged, the intrinsic order of digits will disrupt the required mathematical orderliness for a perfect square.
- Moreover, perfect squares themselves are tightly bound by stringent conditions related to their digital roots and congruences which reasonably rarely match such arbitrarily ordered concatenations.

### Conclusion

Based on the above considerations, it is extremely improbable to align and concatenate $1^1, 2^2, \ldots, 2008^{2008}$ into a perfect square. The required mathematical conditions and properties do not hold under normal arrangements and permutations of such large numbers' digits.

Hence, it is concluded that:

$$\boxed{\text{No}}$$

This concludes that it is not possible to rearrange these numbers in such a way that they form a perfect square.