Proper notation of doubly charged ions
TL;DR: $\ce{He^2+}$ is the only preferred notation.
Notations $\ce{He^{++}},$ $\ce{He^{+2}}$ or $\ce{He^{1+}}$ are obsolete and should be avoided.



From IUPAC “Green Book” [1, p. 49], section 2.10.1 Other symbols and conventions in chemistry, subsection (i) The symbols for the chemical elements:


  The ionic charge number is denoted by a right superscript, by the sign alone when the charge number is equal to plus one or minus one.
  
  $$
\begin{array}{lll}
\textit{Examples} &amp; \ce{Na+} &amp; \text{a sodium positive ion (cation)} \\
  &amp; \ce{^{79}Br−} &amp; \text{a bromine-79 negative ion (anion, bromide ion)} \\
  &amp; \ce{Al^3+}~\text{or}~\ce{Al^{+3}} &amp; \text{aluminium triply positive ion} \\
  &amp; \ce{3 S^2-}~\text{or}~\ce{3 S^{-2}} &amp; \text{three sulfur doubly negative ions (sulﬁde ions)} 
\end{array}
$$
  
  $\ce{Al^3+}$ is commonly used in chemistry and recommended by [74]. The forms $\ce{Al^{+3}}$ and $\ce{S^{−2}},$ although widely used, are obsolete [74], as well as the old notation $\ce{Al^{+++}},$ $\ce{S^=},$ and $\ce{S^{−−}}.$


For the historical perspective, see the corresponding article by Jensen [2]:


  In contrast, the German chemist, Walther Nernst, in his equally inﬂuential 1893 textbook of theoretical chemistry, chose to place an appropriate number of superscripted $+$ or $−$ signs directly above the ion’s atomic symbol (4), a practice which was soon modified by placing them instead to the immediate right of the symbol, as in $\ce{Ba^{++}}$ and $\ce{PO4^{---}}$(5).
  
  The IUPAC guide to Quantities, Units and Symbols claims that yet a third “algebraic” method of indicating ionic charges was also used in the past in which the charge preceded the numerical value, as in $\ce{Ba^{+2}}$ and 
  $\ce{PO4^{-3}},$ even though this particular sequence of symbols was originally intended to represent the inherent sign of a number or exponent and not the number of signs (6). However, inspection of nearly three dozen general, inorganic, and analytical textbooks, spanning the period 1909–1975, revealed that the vast majority employed the modified Nernst notation, with a smaller number — mostly of European or Russian origin — using the Ostwald notation instead. Rather surprisingly, very few examples of texts using the algebraic notation could be found, all of them post-1970 (7).
  
  Since at least the 1950s IUPAC has ruled that ionic charges or “charge numbers,” as they are now officially called, should be written instead with the number preceding the charge sign, as in $\ce{Ba^2+}$ and $\ce{PO4^3-}$ 
  (6, 8, 9). There are several reasons for this decision. It is more concise than the typographically inelegant Nernst approach and more physically meaningful than the Ostwald notation. Unlike the algebraic notation, it 
  avoids confusion with the conventional symbolism for inherently positive and negative numbers and maintains consistency in how we count physical entities. 


References


IUPAC “Green Book” Quantities, Units, and Symbols in Physical Chemistry, 3rd ed.; Cohen, R. E., Mills, I., Eds.; IUPAC Recommendations; RSC Pub: Cambridge, UK, 2007. ISBN 978-0-85404-433-7. (PDF)  
Jensen, W. B. The Proper Writing of Ionic Charges. J. Chem. Educ. 2012, 89 (8), 1084–1085. DOI: 10.1021/ed2001335.