Let $p_1 = 2012$ and $p_n = 2012^{p_{n-1}}$ for $n > 1$. Find the largest integer $k$ such that $p_{2012}- p_{2011}$ is divisible by $2011^k$.
1. We start with the given sequence \( p_1 = 2012 \) and \( p_n = 2012^{p_{n-1}} \) for \( n > 1 \). We need to find the largest integer \( k \) such that \( p_{2012} - p_{2011} \) is divisible by \( 2011^k \).

2. First, we express \( p_n \) in a more convenient form. Notice that:
   \[
   p_2 = 2012^{2012}, \quad p_3 = 2012^{2012^{2012}}, \quad \text{and so on.}
   \]
   By induction, we can show that:
   \[
   p_n = 2012^{2012^{2012^{\cdots^{2012}}}} \quad \text{(with \( n-1 \) exponents of 2012)}
   \]

3. Specifically, for \( n = 2012 \):
   \[
   p_{2012} = 2012^{2012^{2012^{\cdots^{2012}}}} \quad \text{(with 2011 exponents of 2012)}
   \]
   and for \( n = 2011 \):
   \[
   p_{2011} = 2012^{2012^{2012^{\cdots^{2012}}}} \quad \text{(with 2010 exponents of 2012)}
   \]

4. We need to find \( p_{2012} - p_{2011} \):
   \[
   p_{2012} - p_{2011} = 2012^{2012^{2012^{\cdots^{2012}}}} - 2012^{2012^{2012^{\cdots^{2012}}}}
   \]
   Let \( x = 2012^{2012^{2012^{\cdots^{2012}}}} \) (with 2010 exponents of 2012), then:
   \[
   p_{2012} - p_{2011} = 2012^{2012^x} - 2012^x
   \]
   Factor out \( 2012^x \):
   \[
   p_{2012} - p_{2011} = 2012^x (2012^{2012} - 1)
   \]

5. To find the largest \( k \) such that \( 2011^k \) divides \( p_{2012} - p_{2011} \), we use the \( v_p \) (p-adic valuation) function. We need to evaluate \( v_{2011}(2012^x (2012^{2012} - 1)) \).

6. Since \( 2011 \) is a prime number, we use the Lifting The Exponent (LTE) lemma. The LTE lemma states that for \( a \equiv b \not\equiv 0 \pmod{p} \) and \( n \) a positive integer:
   \[
   v_p(a^n - b^n) = v_p(a - b) + v_p(n)
   \]
   Here, \( a = 2012 \), \( b = 1 \), and \( n = 2012 \):
   \[
   v_{2011}(2012^{2012} - 1) = v_{2011}(2012 - 1) + v_{2011}(2012)
   \]
   Since \( 2012 \equiv 1 \pmod{2011} \), we have:
   \[
   v_{2011}(2012 - 1) = v_{2011}(1) = 0
   \]
   and:
   \[
   v_{2011}(2012) = 1
   \]
   Therefore:
   \[
   v_{2011}(2012^{2012} - 1) = 0 + 1 = 1
   \]

7. Finally, since \( 2012^x \) does not contribute any additional factors of \( 2011 \):
   \[
   v_{2011}(p_{2012} - p_{2011}) = v_{2011}(2012^x) + v_{2011}(2012^{2012} - 1) = 0 + 1 = 1
   \]

Conclusion:
The largest integer \( k \) such that \( p_{2012} - p_{2011} \) is divisible by \( 2011^k \) is \( \boxed{1} \).