Can you get addicted from using topical or local anaesthetics to ease pain? I've been wondering if there was such a thing as developing some kind of physical or psychological dependence on things that relieve pain. While we often hear about opioid addiction, which is primarily physical, I wonder if someone could psychologically be hooked on using something like lidocaine to numb every bruise or injury they get, just to numb the pain.

I did hear that cocaine was once used as an anaesthetic, but it was also physically addicting.
This makes me wonder if physical withdrawal and hunger cravings are similar.
Short answer
Basing an answer strictly on the DSM-V criteria, local anesthetics may, purely hypothetically, be associated with a mild form of addiction. But realistically, I think it is unlikely that a person (ab)using a topical anesthetic will ever be medically diagnosed as having a substance abuse disorder.  

Background
I think the question hinges on the definition of [drug] addiction. The APA addresses addiction in DSM-V by the following criteria (entered in shortened form for brevity):


  A minimum of 2-3 criteria is required for a mild substance use
  disorder diagnosis, while 4-5 is moderate, and 6-7 is severe [...]
  
  
  Taking the opioid in larger amounts and for longer than intended
  Wanting to cut down or quit but not being able to do it
  Spending a lot of time obtaining the opioid
  Craving or a strong desire to use opioids
  Repeatedly unable to carry out major obligations at work, school, or home [...]
  Continued use despite [...] social or interpersonal problems [...]
  Stopping or reducing important social, occupational, or recreational activities [...]
  Recurrent use of opioids in physically hazardous situations
  Consistent use of opioids despite [...] physical or psychological difficulties [...]
  [Development of t]olerance [...]
  [Development of w]ithdrawal [symptoms]
  


In contrast to opiates, lidocaine has no rewarding or addictive properties. As a result, lidocaine can be used systemically and on a long-term basis for the treatment of chronic pain (Yousefshahi et al., 2017). In fact, lidocaine, has equally effective anelgesic effects in people addicted to opioids as in normal controls (Hashemian et al., 2013). Because lidocaine is a local anesthetic without central effects, and taken together with above findings, we can safely rule out the occurrence all of the criteria except, perhaps, the first three. Those first three may be, at least hypothetically, possible when topical or systemic application relieves some sort of untolerable pain. Given that meeting 3 criteria meets the definition of a mild substance abuse disorder (mild addiction), I wouldn't rule addiction to topical anesthetics out completely. It does seem unlikely, though.  

Cocaine has strong addicitve effects, because of it's pronounced effects in the central nervous system - it is associated with strong feelings of reward and euphoria. It rapidly induces tolerance as well as dependence; a school book example of a substance of abuse. Lidocaine is a cocaine derivative that still exhibits its anelgesic properties, without its psychoactive component (Small, 2009).    

References
- Hashemian et al., J Emerg Trauma Shock (2014); 7(4): 301–4
- Small, Subst Use Misuse (2009); 44(13): 1888-98
- Yousefshahi et al., Anesth Pain Med (2017); 7(3): e44732