What additional aircraft equipment is required to fly a US registered airplane in Europe?
An overview of ATC related avionics can be found on the Eurocontrol website. There are a couple of things changing in 2015.

The list below includes change to ATC related avionics requirements in 2015. 

In addition other avionic requirements not related to ATC are imposed, but that is outside my field of expertise. 



Surveillance

Commission Implementing Rule (IR) (EU) No 1207/2011

Commission Implementing Rule (IR) (EU) No 1028/2014 (Amendment of 1207/2011)

ADS-B

ADS-B (1090ES) will be mandated  from 7 January 2017 for aircraft with MTOM > 5700 kg and Max cruise TAS > 250 Kts with an individual  certificate  of  airworthiness  first  issued on or after 8 June 2016. For aircraft certified before 8 June 2016 with the same MTOM and max cruise speed class ADS-B will be mandated after 7 June 2020 . 

Applies to all IFR flights in the European Civil Aviation Conference (ECAC) airspace, excluding oversea territories. 

Mode-S Elementary Surveillance (ELS) 

Mode S ELS will be mandated for all aircraft operating according to Instrument Flight Rules from 8 January 2015

Applies to all IFR flights in the ECAC airspace, excluding oversea territories. 

Mode-S Enhanced Surveillance (EHS) 

Mode S EHS will be mandated for new aircraft with MTOM > 5700 kg and Max cruise TAS > 250 Kts from 8 January 2015. For existing aircraft in the same MTOM and max cruise speed class Mode S EHS will be mandated after 7 December 2017.

Applies to all IFR flights in the ECAC airspace, excluding oversea territories. 



Communication

Commission Implementing Rule (IR)(EU) No 29/2009

Controller-Pilot Data Link Communications (CPDLC)

CPDLC (VDL Mode 2) is mandated above FL285 from 5 February 2015

Applies to all IFR flights in the ECAC airspace, excluding oversea territories.



Safety Enhancements

Commission Implementing Rule (IR) (EU) No 1332/2011

TCAS

From 1 December 2015 TCAS II version 7.1  will be mandated for all aircraft build before March 2012 with MTOM > 5700 kg and Max cruise TAS > 250 Kts operating in EU airspace. Aircraft first certified after 1 March 2012 already need to operate with TCAS II version 7.1.

Outside EU airspace, but within ECAC airspace, TCAS version 7.0 is still allowed.