In RSA Encryption, Can the public exponent e > m (modulus) ? and can we choose any public key without paying attention to the conditions?

Question

In RSA ,the encryption, Can we choose the public exponent (e) greater than m (modulus) or e > φ(n) ? I wonder about choosing public key exponents (e) because the most information on the internet or articles has generated public keys from these 2 conditions :

1. 1 < e < phi(n)

2. e < m

So is it possible that we can choose to use e>m or e > phi(n) for the publickey encryption. And If e>m will there be any problems ? (other effects than make the calculation slow)